The Eleventh Premise – Types of Infinity and the Dispute over Potential and Accidental Infinity | Part I Chapter 73 (18) | Guide for the Perplexed 173 (Auto Translated)

Premise 11 of the Kalam: The Negation of the Infinite in Every Respect

The Position of This Premise in the Overall Structure

We are studying the eleventh premise, yes, of the 12 premises of the Kalam. As we recall, all these premises are premises concerning the foundations of existence in physics, and one could say epistemology and metaphysics, but fundamentally this is primarily physics, upon which they will prove the four sought-after conclusions—which are the matter of God’s existence, His unity, the negation of corporeality, and the creation of the world.

The direction of the Kalam is of course from the beginning—they always start with the creation of the world, and from this arrive at God’s existence, and from this to the two important attributes of God: that there is only one, and that He is not a body.

Why am I telling you this? So that we know where we are in the world. And also because later, when we reach the next chapter, we will see that these two premises—the tenth and the eleventh—are actually the most important premises. That is, they are, as it were, the epistemological and metaphysical conclusions that follow from all the other premises, as he explained, for example, in the tenth premise at great length, how this is based on all the other premises. And therefore upon them, more directly, the proof is built, or most of the main good proofs for the creation of the world are built upon these premises.

The Rambam himself, regarding the previous premise—the premise of possibility, the tajwiz—the Rambam himself actually wants to prove, or of course does not think, or as is known does not think that one can prove the creation of the world, but his statement that one can approach a proof for the creation of the world, he bases essentially on the previous premise—the tenth premise—as he indicates here at the beginning.

The Nature of Premise 11: The Metaphysics of the Creation of the World

Premise 11 is more or less the essence of the question of the creation of the world, yes? It is not just a kind of metaphysical or physical foundational assumption upon which the creation of the world is built, but it is more or less the metaphysics of the question of the eternity and creation of the world.

And therefore, if Premise 11 is correct, as he will say at the end of this premise, then we have already proven the creation of the world. This itself is, as it were, something—it is not exactly to say the creation of the world, but it is to say its metaphysical foundation. But because the Rambam thinks this is not so simple, it is not so simple to prove the creation of the world.

And therefore the essence of this premise deals with defining, with positing this premise—this assumption upon which the Kalam truly builds most of the proofs, or a large part of their proofs for the creation of the world. And one can say that most of it is based or built around the idea of distinguishing between types of infinity, as we shall see, and explaining why there is actually a question here.

The Text of the Premise: The Negation of the Infinite “in Every Respect”

He says thus: The eleventh premise is their statement that the existence of that which has no end—that which has no end, that which has no limit, that which is infinite, as we call it, and unlimited—is false in every respect, yes?

False, that is, impossible, absurd, cannot be. False is a good word—false in every respect, meaning in every type of infinity, as he will explain in a moment.

So this is their premise, this is the statement of their premise. Infinity cannot be, there cannot exist in the world an existence of something limitless, infinite.

The Main Innovation: “In Every Respect”

And they do not distinguish—that is to say, this is their main innovation, as he will explain, this is this word, this line “in every respect,” yes, or in every aspect, in every manner of infinity, yes. Not in every respect of anything, but in no manner, in no type of infinity, yes. There is no type of infinity that exists, yes.

It may be that theoretically or mathematically we can think about it, but existentially, yes, in terms of existence—existence, yes, this is the word existence, yes, this is very important. In thought, everyone agrees that one can count numbers to infinity, this is not a debate. But existence, yes—so this word existence is also important.

The phrase “has no end” is the general principle, but their main statement is that its existence, that it exists also outside of thought, outside of knowledge, that something without end exists—is impossible. And then the word is important “in every respect”—in whatever manner, every type of infinity is impossible.

The Connection to an Eternal World

We must remember, we can preface this, that an eternal world is nevertheless an infinite world in a certain sense, yes? That is, infinite in time, yes? That it never began and will never end. So it is infinite in a certain sense, which is the sense of time.

And the Rambam will actually explain here, and we will also reach this in the following chapters, or not so much, but mainly in the first premises of the second part, to explain the types of infinity—what are actually the proofs that they do not exist and which of them can exist and so forth.

And here what he says is that the Kalam thinks that infinity cannot exist in any manner. No type of infinity can exist.

Explanation of the Premise: The Agreed-Upon Types of Infinity

And the explanation of this—what is the explanation? That is, what is the explanation of, I think this is an interpretation of the phrase “in every respect”—how many types of respect, how many types of infinity are there.

He says thus, and he will count four types, then he will count four types again, and I am not sure they are the same four, but let us see, perhaps there is something strange here.

What Has Already Been Proven by Demonstrative Proof

The explanation of this is thus, for it has already been clarified—we have already proven. “Has been clarified” is not a sufficiently precise translation of “lubrihan.” “Lubrihan” does not sufficiently distinguish between the various verbs that pertain to proof and clarification.

We know that among technical Aristotelians there is a great distinction between demonstrative proof—what the Rambam always calls apodictic proof (mofet)—and various arguments and various proofs of lesser types.

Here it says proven, yes, and Schwarz is careful about the correct translation of such things, and he translates that it was proven by apodictic proof, yes. This is very important, because these are proven things, yes—proven insofar as physics is proven. But the Rambam takes Aristotelian physics as more or less proven. True, there is the doubt he raised at the end of the previous part, that perhaps this is not necessary in the sense that there could not have been another world—in our world this is necessary, or is it logically necessary—but in any case this is called apodictic proof.

“Has been clarified” is a word that cannot be so, and he now counts three types of infinity that he will also count in the introduction to the second part. There too he does not give the premises, he says it has already been clarified, meaning you need to read Aristotle’s Physics—I think in the sixth book—there Aristotle proves in logical forms the impossibility, the impossibilities of these infinities.

First Type: One Infinite Magnitude

Existence—so what has been clarified? This is agreed upon actually, for they only add to the impossibility of the infinite, so whatever is also agreed upon, certainly this is agreed upon.

It has already been clarified that there is no existence of one body that has no end. Here it says “one body,” but it may be that this is not necessarily a body, and it is not certain that in the original it says body. Of course magnitude—what is called magnitude in English—is apparently a property of a body, so one can translate body, but in the original it actually says only magnitude, magnitude—something that has magnitude of course.

But the problem is that magnitude is by definition something limited. There cannot be a magnitude that has no end. If we describe one body that is infinite in its magnitude, has no boundary, has no sides, has no outside and so forth—this is impossible.

So this is one thing, this is the first, simplest type to prove. It may be that not everyone agrees with this, but Aristotle thinks, and we need to arrive at seeing the proofs for this, I do not know them today, but Aristotle says that it is impossible for there to be something of infinite magnitude. This is one thing.

Now, this is the simplest thing, this is the simplest type, and indeed it is hard even to imagine this. One can understand where the impossibility comes from, but one needs to search for the more rigorous proof for this. But this is simple.

Second Type: Infinite Things of Finite Magnitude—Existing Together in Time

Now, there is another thing. Not only one infinite thing, but also a second thing: or the existence of bodies, yes, or as I said, one needs to read here “magnitudes”—things possessing magnitude—infinite in their number, yes?

That is, each of them is indeed finite—we have many things, or infinite things, each of which possesses finite magnitude, but there are infinitely many of them, yes? And even though each of them, its body or its magnitude is finite.

So this is a second thing that we agree upon, that it is necessary, it has already been proven by apodictic proof, that this type of infinity cannot exist, yes? This is not—we imagine this because infinite space is needed for this, but one must remember that Aristotle does not believe in place, yes? So the problem is not place. The problem is again, that this somehow adds up to an infinite magnitude that is incoherent, impossible, if I remember correctly.

So this is a second thing, yes? That there cannot be two things, or many things, yes? Infinite things, each of which possesses finite magnitude, but it is in the end—number, one can say, an infinite number of finite things, of finite things.

The Decisive Condition: Simultaneous Existence

Okay? Now, and here there is an important condition, and we will see later that the Kalam disputes this condition, yes? And on the condition, yes—now, when we say that infinite things of finite magnitude cannot exist, we mean this, Aristotle means this with a very important condition: on the condition that each one, that these that have no end, exist together in time, yes?

Because the problem, yes, this is actually—I am saying an argument thus from my perspective.

Second Type: An Infinite Number of Finite Things Existing Simultaneously

This is actually the problem. And therefore this problem exists only when all these things, all these infinite things, exist at one time. So at one time, as it were, it cannot be. I will not say there is no place to put them, because there is no place in the world. But it cannot be that such a list of objects, of things possessing infinite magnitude, should exist.

This is an important warning, because later we will see in the fourth division, that if they do not exist at the same time, then Aristotle says there is no proof, and indeed it may be, it may be that it must be, but certainly it may be that if today there is one thing of finite magnitude, tomorrow there is another thing, and thus this tomorrow continues and extends to infinity, this is not a problem, because they do not exist together so that we get one cumulative magnitude of infinity. This is more or less the matter. So these are two more types of infinity that are impossible.

Third Type: An Infinite Chain of Causes—Metaphysical Infinity

The Transition from Physical Infinity to Metaphysical Infinity

And thus there is a third type of impossible infinity, and this is actually metaphysical infinity, not physical. Until now we spoke about a physical problem, as it were, a problem of magnitude, magnitude—infinite magnitude cannot exist. And regarding this there is a third thing, and likewise the existence of causes that have no end, and regarding this third thing there is certainly no dispute, this is perhaps the strongest of all, but for this one needs a proof of another type.

In any case, and likewise the existence of causes that have no end, causes where each one is a cause for this and one is a cause for that is false. This is called infinite causality. That is to say, and he explains what I mean, causes that have no end. That is to say that there should be a thing that is a cause for another matter, and for that thing another cause, and for the cause a cause, and so it has no end. Until there should be movers, whether countable things, without end, infinitely many existing in actuality, whether bodies or separate entities, but some of them are causes for some of them.

Bodies and Separate Entities

Here he explicitly writes the word bodies, that is, until now, and therefore perhaps Ibn Tibbon translated thus, before this we spoke about bodily things, indeed magnitude is a property of bodily things, bodily things have no magnitude. But now we are adding another type, more abstract, more distant from the first thing or the simplest thing, now we are moving away, we are saying, there is another type of infinity that cannot exist, and this is that there should be infinite causes.

The Explanatory Problem in an Infinite Chain of Causes

If I remember correctly, the main problem with this is an explanatory problem. It is not that we have no place, as it were not that we have place, this is indeed incoherent, but it is incoherent because an explanatory problem emerges. A cause is an explanation, why something exists, and if these explanations go to infinity, then it turns out there is no explanation, that the explanation must always be something that ends at some point, so that we can say, okay, and this is the cause. And therefore, causes without end are simply not a cause. This is the important point.

The Decisive Distinction: Efficient Cause versus True Cause

An Efficient (Historical) Cause Is Not a True Cause

Because cause here, the intention is a true cause, not a historical cause, not a cause of, what do you call it? of this thing caused that historically, because this may actually somehow turn out, one needs to threaten with this, but it does turn out according to the system of premises that there is such a thing, for there are changes, and every change is connected to the change that comes after it, and this continues indeed to infinity. But here the intention is true causes.

And true causes cannot exist together, because a cause, and now there is another important thing, causes are things that exist at the same time, for there is a simple law here.

The Efficient Cause Does Not Fall Under This Law

Again, this does not apply to every cause. The efficient cause, what we call, this does not apply, this applies to the other types of causes perhaps, but not to the efficient cause, which is the historical cause, to which we are accustomed only. One must remember that in the type of causes to which we are accustomed, it may be that this is not true at all. I say in the notes, one needs to check them, but thus it turns out, I think, yes?

The Example of the Table: A Historical Cause Does Not Explain the Present

For the causes we are accustomed to are all historical causes. We say, why is my table located where it is located? Because yesterday someone put it there, and yesterday no longer exists, and the movement that put it here yesterday no longer exists, and therefore there is no problem at all that causes of this type can continue to infinity, because they do not exist together in time, yes?

And also, it is true that there is an explanatory problem here, yes? Because this is a problem with this type of cause, yes? Because this cause does not explain why the table is here. It only tells us how it got here and does not explain why, at all that this required a cause of the type of Form and Matter and Purpose and types of causes of this type, which would be true causes at all.

True Causes Must Exist Simultaneously with the Effect

And now those causes, the causes that truly explain the thing, they always exist, yes? About those causes we say, it cannot be that the cause went upon it and disappeared, yes? Not as Avicenna thinks that in the sense he understands cause, a physical cause, which can, as a person can build a house and now leave. This actually teaches us that the person is not a cause for the house in a true manner.

A cause that is truly an explanation for how, for the existence of the thing or for how the thing is now and so forth, must be a cause that exists now, because otherwise, why is it still, it is still now. And therefore, yes, one who believes that things of yesterday are an explanation for now, still needs to add some cause, and besides there is some property of nature or time or something that says that things of yesterday are still today.

Cause in Actuality versus Cause in Potentiality

Okay, so this is the cause, yes? In any case, this type of causes that exist, yes? Existing in actuality, yes? Because things in potentiality are not causes for anything. This is another explanation, yes? Another important thing, yes? A cause that acts, a thing that does something, which means it exists in actuality, it is not an idea, it is not a possibility. A possibility does not act. A possibility is a thing from which something is made or to which something is made, but it is not yet a thing that acts, not an active thing, not a thing that is in actuality.

Therefore, a cause that is in actuality must be in actuality, and a cause must exist simultaneously with the thing for which it is a cause. And here we have the law of preventing an infinite chain of causes, because an infinite chain of causes are not causes. If I remember, this is the cause. There may be other proofs for this thing, but this is also a necessary negation.

Summary: The Three Types of Impossible Infinity

Causes that have no end is false, this is a third type of negation of infinity existing at once, yes? This is important, this understanding that I added is important only generally for understanding all types of proofs for this, yes?

The Relevance to the Debate with the Kalam on the Eternity of the World

The Kalam: The Eternity of the World Requires an Infinite Chain of Causes

For the Kalam, here exactly is the point of the debate in this premise, because the Kalam says exactly about this, if you say creation of the world, eternity of the world, you are saying causes without end, yes? And for this we need to enter into this examination of what is meant by causes without end.

That is, indeed, because they do not have the distinction between these types of infinity, it turns out for them, as we are saying all the time, it turns out for them a material historical understanding of causes, and they do not believe in causality at all, so they believe only in God’s will and so forth, so they do not have the problems at all.

The Aristotelian Answer: Distinguishing Between Types of Causality

Whereas the attempt of the Aristotelian understanding that allows us the distinction between these types of causality, allows us to say that the world continues infinitely in time, but it is not that every prior thing is a cause for the thing that comes after it, because this would indeed be an infinite chain of causes that does not provide an explanation.

And therefore indeed the Aristotelian world, despite being infinite, does need an explanation, at least an explanation for somehow, that is for its motions. It is not clear that Aristotle believes in explanations for its very existence, but this is another question.

An Internal Problem in the Kalam Position

I think there is something internal here, yes? That there is no, as if, if you do not believe in causality, then where does this law come from that there cannot be infinite cause? But in any case, generally this is still agreed upon on both sides here that all infinity of the type, all three of these types of infinity, they cannot exist.

Summary: The Natural Self-Evident Order

Up to here three types of infinity. They have been clarified, yes they have been clarified, that we, they are impossible, they are impossible. And the Rambam concludes, and this is the natural order, that is, the negation of these three types of infinity, this is the natural self-evident order, yes?

All these are, as we, as we learned, how do I say? As we learned at the beginning of the second part, all the wisdom of physics is based on these negations, and therefore this is the natural self-evident order which is demonstrated by apodictic proof, it is prevented from a kind that has no end.

Yes, all this order, this order, the course of the basic order of the world, that impossible things, sorry, that infinite things cannot exist in these three aspects, these are necessary things, and the wisdom of nature is built upon them, and this is demonstrated by apodictic proof, all this is agreed upon, yes?

Of course not agreed upon by the Kalam specifically, yes? Because the Kalam do not believe in natural order. I think there is something internal here, yes? That there is no, as if, if you do not believe in causality, then where does this law come from that there cannot be infinite cause? But in any case, generally this is still agreed upon on both sides here that all infinity of the type, all three of these types of infinity, they cannot exist.

However. But there is here another…

Summary: The Three Types of Infinity Agreed Upon as Impossible

Generally this is still agreed upon on both sides here, that all infinity, all three of these types of infinity, they cannot exist. However, but there is here another one, another type of infinity, upon which there is no natural self-evident order by apodictic proof.

Apparently actually, yes, as he will say, he thinks there is no apodictic proof for either side, and this is exactly the sense in which there is no apodictic proof for the creation of the world. That is, and therefore also for this question there is no apodictic proof. There is deep inquiry, there is aporia, I think, there is doubt, and this doubt goes nowhere. There is doubt here.

The Fourth Type: Infinity in Potentiality or by Accident

Chunk 2 – English Translation

However, what is the distinction? What is the other type of infinity? An existent that has no end, potentially or accidentally. Yes, these two types that are listed here are one, but the Rambam also later goes on to count them as three and one, but they’re not exactly three and one. An existent that has no end, not in one magnitude, not in many magnitudes, not in causes that are an infinite chain, but potentially.

Infinity in Potentiality — Not in Actuality

Not in actuality. We remember, in actuality means to exist at the same time, and that’s a problem. But potentially means it’s only potential, it doesn’t exist, it hasn’t been actualized, there’s only the thing, this Aristotelian secret called being in potentiality, which is not to exist, it’s some kind of middle ground between existing and not existing. But at least it doesn’t exist in the sense that it has a problem of infinite magnitude.

Infinity Accidentally — One After Another

Or accidentally, what does accidentally mean? Accidentally is a somewhat strange word. Remember that a moment ago he said natural and essential, this is the natural essential order, meaning essential as opposed to accidental and natural as opposed to actual, I don’t know, but this is the essential order.

Infinity accidentally is a word that the Rambam is going to explain, he’s going to talk about infinity of one after another, that’s the more precise word for it. But accidentally means a kind of, it’s not that the thing is infinite, but rather that one can count infinity here, as if from the side, as if accidentally infinity emerges here, but there’s no infinity within the essence of the thing itself, right? It’s not that it has infinite power, one could say, or something like that.

Infinity in Potentiality — The Infinite Division of Body and Time

So these things, so like this, if we’re talking about infinity potentially, or accidentally, the Rambam says like this, part of this thing, meaning potentially, that’s what he’s going to mean, and about this there’s also a dispute with the Kalam, which I don’t know why he doesn’t make present here, one needs to examine more carefully the account of this topic.

Part of it, meaning one of them, some of them, that whose existence has already been demonstrated by proof. There’s another type of infinity, and the Rambam claims there’s a proof here, there’s a type of infinity that we actually know exists. We, meaning the Aristotelians, those who believe in the nature of reality, only the world doesn’t believe in this.

The Proof: Body and Time Divide Infinitely in Potentiality

As has been demonstrated by proof, body will divide without end potentially, and time will divide without end. It’s interesting that he doesn’t write about time without end potentially, but he still means potentially, in this sense.

After all, we learned in the first propositions here that there’s an ancient dispute essentially, but a major dispute here, whether time and place, time and objects, things are continuous or discrete? Are they Discrete or Continuous?

Continuity Versus Atomism

And for those who say they’re continuous, that’s equivalent to saying, it’s saying in another word that they’re infinite, that one can divide them infinitely, right? That’s the answer to Zeno’s paradoxes and such things, right? We can divide the… every physical thing, every body, we can divide it infinitely, right? And therefore it maintains continuity. Every part in it touches every part in it, right? There’s no, there’s no real separation within the parts.

In contrast to this is the doctrine of atomism, which, yes, claims that there aren’t really things, there are only things adjacent to each other, but they’re never continuous in their existence, right? There’s always, if we divide them, we’ll reach an indivisible part, which is the atom, right? The atom, an indivisible part.

The Proof of Continuity Is Connected to the Concept of Infinity

But this has already been proven, and the proofs for this are connected to the concept of infinity, right? Again, one needs to examine in the Physics, where he explains these things, if one reads. But these things are connected to problems of infinity, to the paradoxes I mentioned and such, and we, we have proof that body divides infinitely.

Time Divides Infinitely Because Motion Divides Infinitely

And therefore, as the Rambam already taught us in the second and third propositions, here it goes together with this that time also divides infinitely, right? Because after all, motion is the movement of bodies or change of bodies, and therefore if body divides infinitely, therefore also its motion and the time that is the number of motion, right? Aristotle’s statement that time is the number of motion, time is the thing that measures or that we count the before and after of motions, so it also divides infinitely.

Therefore because we don’t believe in atomism, according to all the things he mentioned, we have demonstrative proof that body divides infinitely.

Imaginary Lines Versus Real Division

And similar to this we already spoke earlier about the ideas of lines that go to infinity, but there those can be imaginary lines, they don’t have to be lines that can exist, but in the sense that we can divide body infinitely this has been proven according to Aristotle, it’s been proven that this exists.

The Aristotelian Answer to Zeno’s Paradox: Infinity in Potentiality and Not in Actuality

But if one asks Aristotle, and here, this is Aristotle’s precise answer to Zeno’s paradox. Zeno asks him, okay, so how can you traverse an infinite distance in order to reach anything? After all, everything divides forever.

Aristotle says, no, you didn’t understand, this is infinity in potentiality and not in actuality. The infinity that things divide into, this doesn’t mean that we can divide, this also doesn’t mean that we can divide, it could be that we don’t have a knife sharp enough to cut anything infinitely, and this also certainly doesn’t mean that we’ve already divided anything infinitely.

Meaning, the thing as it is in itself is not divided infinitely, it’s limited, it’s finite, it has an end. When I say it’s continuous, I’m only saying that it can be divided infinitely, right? It has possibility, it has potentiality, and here the Aristotelian patent of potentiality, once again saves the day, he says there’s infinite potentiality, and therefore we think about it, and the true structure of matter, of material things, is always that they’re infinite, that they can be divided infinitely, but this doesn’t mean that it’s actually divided infinitely.

Therefore most of the paradoxes of infinity, which the Aristotelians agree about that actual infinity cannot exist, are not problems.

The Position of the Mutakallimun: Denial of Infinity Even in Potentiality

And this has been demonstrated by proof, and here he’s as if, the Kalam is completely wrong. They claim that even in this sense there’s no infinity, and this is connected, as we already learned in the previous proposition, I think this is connected to the denial of the truth of infinity. One who doesn’t enter into these distinctions between types of infinity, he really needs to deny the continuity of matter, and for similar reasons.

Okay? So but, okay, but this isn’t our problem now, this isn’t the main problem for us.

The Fifth Type: Infinity Accidentally — One After Another — The Undecided Question

But part of it, the second part is actually the problem. Part of it, that which is subject to investigation. There’s another part of infinity that is, as he defined a moment ago potentially, or accidentally, and then he already spoke about potentially, now he’ll speak about accidentally.

So actually according to the count here this is the fifth type of infinity, and this is that which is subject to investigation. There’s room for investigation here. It could be that this division is potentiality and accidental, or not really a real division. Maybe one can also call this infinity in potentiality if one wants, or therefore both can perhaps be called potentially or accidentally, not sure about this, one needs to look.

What Is the Aporia?

There’s room for investigation here. Room for investigation is what I call aporia. There’s a question here that the Rambam never decides.

And he, but this is the question, this is actually what the dispute here is about. An existent that has no end, in coming one after another. Not infinity all at once, and not infinity of something that can be divided theoretically, or potentially into infinite parts, but rather that each time there’s something new, that comes one after another. After the previous one disappears, it goes, now there’s something new, but this sequence, this chain, which is not a causal chain, but continues infinitely. That’s the question.

And this is what is called that which has no end accidentally. We can call this without end accidentally, right? Not in itself, not that there’s a thing that has no end, and also not a sequence, not a list of things that are infinite, that is infinite, but infinity does exist here, but accidentally.

And it will be one thing after the absence of one thing, and that other after the absence of a third other, and so on without end, right? And so it continues forever.

This Is Deep Investigation — Undecided Perplexity

And in this is the very deep investigation. Here the Rambam says this is deep investigation and he means something undecided, right? Deep investigation, there are good questions, there are good arguments for the different sides, there are reasons to think this and arguments to think that, but it’s not deep, difficult, there’s perplexity here. I think the word perplexity is the most precise word here. There’s perplexity here, there’s a problem here.

The Argument for the Eternity of the World: Time Is Without End

And the Rambam tells us, and whoever boasted that he demonstrated by proof the eternity of the world, after all there are claims, and there really are claims also for this side, if we get to the arguments in favor of the eternity of the world, we’ll see that it’s very difficult to describe the beginning of motion, the first motion. Actually the belief in creation says that at a certain stage we agree to think, but one who wants to understand, so there are, there are actually proofs that motion must be infinite, it must be.

So in any case, the Rambam doesn’t agree with this, so it’s written here whoever boasted or whoever claims, it’s not clear that they saw me also whoever boasted. But Aristotle, or as we’ll say in the second part, some of Aristotle’s followers claimed that they have proof for the eternity of the world, and they will say that time is without end, yes, time never ends, and no falsehood will follow from this for him, and this doesn’t contradict his belief, the principle that there’s no infinity, no falsehood will follow from this. Why? Because…

Why Infinity in Time and Accidents Does Not Constitute a Logical Problem

It’s true that there’s infinite time, time didn’t begin and doesn’t end, but all the times are never present all at once. This is the peculiarity of time — that one part of time exists and a second part is already gone, and a third part is going to come. And going to come doesn’t have real existence, and therefore it’s not a problem at all.

This is regarding time. And so also there’s another way that we can describe in the eternity of the world the sequence, the infinite chain: and so accidents come one after another upon matter without end. After all there’s one matter, prime matter, that suffers, it bears all the accidents in the world infinitely. Time is only the description of the change of this matter infinitely. But upon one matter there are infinite accidents — if today thus, and tomorrow thus, and the day after thus, and yesterday thus, and the day before thus, and so on without end, no falsehood will follow for him, and there’s no falsehood here.

Meaning, the Rambam here is teaching this Kalamist, who is certain that Aristotle simply missed something simple — he after all says there’s infinity, and after all it’s known that there’s no infinity. He says: no, you’re not distinguishing enough. Because their not all existing together — because they don’t all exist together, therefore it’s not that there’s a thing here, there’s no actual existence of something infinite. But in coming one after the departure of another — each one comes after the other disappears. And this is something that no proof has come for its impossibility — not something that there’s proof for its impossibility, it’s not clear that this is impossible, there’s no proof. It could be yes, it could be no.

The Two Central Distinctions That the Kalam Denies

However, the Mutakallimun — but these Mutakallimun, the Kalamists — there is no difference for them. They don’t distinguish all these distinctions within the concept of infinity.

First Distinction: Infinity in Magnitude Versus Infinity in Smallness

There is no difference for them between saying that one body exists if you give it an end — the first infinity — or saying that body and time will receive division and will not have an end for it.

And here he skipped three stages. Before this we spoke that there’s one body or many bodies or infinite causes, and all of them are included within “one body exists if you give it an end” apparently. And here the main thing that the Kalam says, and here he mentions explicitly — the Kalam denies this that one can divide. They don’t distinguish between the first three types of infinity, which are all — that one infinity exists all at once, there’s one infinite magnitude in the end, physical or metaphysical magnitude, doesn’t matter — and they don’t distinguish between this and the division of body and time without end for it — the infinity of smallness, not infinity of largeness, but infinity of smallness, this is the opposite, this is infinite small. They don’t accept the difference between this and that. This was one difference that they deny.

Second Distinction: Infinity All at Once Versus Infinity in Sequence

And therefore also the fourth thing they deny. And there is no difference for them between the existence of things infinite in their number arranged together — there’s no difference at all for them between many things being all at once and many things not all at once.

And he gives an example: as if you said, individual people who exist now — there’s no difference between someone who says there are now infinitely many people in the world, or you said, when things came into existence, their number has no end — there were, there were once infinitely many people, but now there aren’t. There were infinitely many people, if you count the whole chain all at once, but this chain never existed all at once. And even though the first ones disappeared one by one — and the Rambam suggests to us exactly: as if you said there is Reuben son of Jacob, and Jacob son of Isaac, and Isaac son of Abraham, and Abraham son of someone whose name we don’t mention, and so on without end. This also for them is false like the first.

The Example of the Chain of Generations — Note on the Translations

This is Ibn Tibbon’s translation, because the Rambam wrote here again Zayd and Sa’id and all kinds of Arabic words that are customary to write for such examples. Schwarz translated David son of Jesse son of Obed son of Boaz, I have no idea why. But in the original it’s written Zayd, who is the son of Umar, who is the son of Khalid, who is the son of Bakr. Well, in any case — these were Jews, Arabs, I don’t know. Why translate, I understand Ibn Tibbon, I don’t understand why they think they’re smarter, I don’t know.

Summary of the Four Parts of Infinity — Difficulty in Structure

In any case, so they don’t distinguish this. And these are four parts of that which has no end that are not equal for them. Which four parts? And now I’m supposed to think like this: the infinity where all exist together, and the infinity where they are one thing existing in it infinite potentiality — not infinite potentiality, but divides potentially infinitely — or, yes, that’s two there, two, one and two. And the third and fourth are: the infinity of existing all at once, and the infinity of not existing all at once.

It’s very strange that at the beginning of the passage the Rambam counted four parts of infinity in a different way. Apparently the distinction is that according to Aristotle the distinction should be like the previous division, and in order to — in order to emphasize the dispute he adopts the division in a different way. But I think maybe there are secrets here and I don’t know them. But and there’s another difficulty here — why, I don’t know, I think there’s a puzzle here, I don’t know what to do.

And this is the last part — whoa, which last part? The fourth part? A moment ago you divided them into four, which are two that are four, and it turns out that the second and fourth part was the dispute. But of course, he has a problem, because the second part — which is time will receive division and place will receive division — this already belongs to the first propositions, which he already spoke about there. Now he’s not talking about this, now he’s only talking about the fourth part. But it’s interesting that all the parts are included in this.

The Connection Between Atomism and the Denial of Infinity

Therefore, as I said, all these propositions are indeed connected. Also this proposition of infinity, despite it being a kind of general metaphysical proposition, it’s also very connected to the matter of atomism. Because one who believes in atomism, part of the reason for this, or part of what this entails, is lack of belief in this distinction between infinity in potentiality and infinity in actuality. And therefore also no belief in infinity all at once and infinity not all at once. And therefore perhaps also connected to the deeper thing, which is no belief in abstract causes, and therefore difficulty in understanding that there’s a thing at all called creation of the world or God’s responsibility for the world without this being a beginning in time. All these things go together.

The Position of the Kalam on the Last Part — Internal Dispute

So now, what do they say about the last part? They think this is impossible exactly, they don’t enter into all this dispute. And we’ll see what they exactly say, or what the Rambam’s summary is of what they say. And they say this is impossible. There’s an intuition here — I’m again describing this from outside, one needs to read what they exactly say, and the following chapters also to enter into their proofs that are based on this. But from outside there’s some very intuitive move here: they claim, everyone understands that infinity cannot be, so how can there be eternity of the world? This is after all infinity, that’s all. They don’t enter — Aristotle says: no, think for a moment, there are types of infinity and so on. In all this they don’t believe, and of course there are real reasons for this dispute.

Now, what do they say about this? So there’s something interesting here. And this is the last part — there’s a dispute among the Kalam how actually — everyone agrees that this is impossible, this infinity, but not everyone agrees how to speak about this, how to prove this, what the proof is.

The Rambam said there are necessary proofs, demonstrative for the denial of the other types of infinity — two or three other types of infinity. But this type, after all Aristotle doesn’t believe in this proof. So what do they say?

Two Approaches Among the Kalam

Some of them will endeavor to verify it — there are some of them who endeavor to verify, to prove (to verify is another word, to prove, many times). To prove, meaning, to prove that such infinity is not possible. Meaning to demonstrate its impossibility in the way they brought in this statement — they want to prove that infinity of this type is not possible, and the Rambam means here what he’s going to write at the end of chapter 74, not sure it’s really at the end but towards the end of chapter 74, where he brings a certain proof for this that there’s infinity, and he claims that this proof itself is based on the lack of distinction between infinity in potentiality and infinity in actuality, and things that are infinity in thought and infinity in reality. So as if, it’s not clear that one even thinks there’s a proof here.

The Argument of “Infinity Greater Than Infinity”

I can go over in one second what’s written there, we’ll get to it there. They claim like this: that if you say there were infinitely many people — and this is interesting because, well, of course, we’ll get to this, we’ll see that there are many Jews, people always, who think in these directions. If you think there were infinitely many people, as he quoted, Jacob son of Isaac son of Abraham son of Terah son of, son of, son of, son of, son of, son of, and it never ends — therefore the Torah begins with these sons that end. The Torah begins with Adam and his name was Seth, and this is the first man, according to, of course, at the beginning of the book, this isn’t our matter. But that there’s a first man, this is exactly the same thing as creation of the world.

But then they claim like this: so infinitely many people have already passed, and then there are also going to be infinitely many more people. So how many infinities are there? How can it be that after all everyone — you have to agree that if now another son is born to me and there are more people in the world, there’s one more, right? And infinitely more are going to be born. And then it turns out there’s an infinity greater than infinity. This seems absurd to me, right? How can it be that there’s already infinity and now we’re increasing it?

The Argument of Absurdity: Infinity Greater Than Infinity

One of the arguments of the Mutakallimun against infinity one after another is like this: suppose there were already infinitely many people in the past, and then there are also going to be infinitely many more people. So how many infinities are there? How can it be? After all everyone, you have to agree that if now another son is born to me, there are more people in the world, there’s one more, right? And infinitely more are going to be born. And then it turns out there’s an infinity greater than infinity.

This seems absurd to me, right? How can it be that there’s already infinity and now we’re increasing it? One of the rules of infinity is something that cannot be increased. At least that’s how one of their arguments goes, it’s an interesting argument.

Maimonides’ Response: The Confusion Between Existent and Non-Existent

And Maimonides argues that this is not a good argument because you are simply confusing between existent things and non-existent things, things that exist only in thought. All this that you are saying, meaning that there aren’t really two infinites here, there is only one infinite and it never actually exists in actuality, then there is no problem at all.

Well, one needs to look at how much the argument works, but this is at least according to Maimonides’ presentation there, later on, that this is not a proof, yes they will consider it one, he will try to moderate, but it’s not really a proof. If this is a proof, it’s a circular proof, based on the same assumptions about the nature of infinity that they assume, but it doesn’t prove anything to someone who doesn’t hold assumptions of this type. Okay? So this is part of the Kalam.

The Second Approach: Negation of the Infinite as a First Intelligible

And some of them say that this is self-evident, and known at the beginning of thought, and does not require proof. So there are Mutakallimun, I don’t know which of them is more sophisticated, who say, no, we don’t need to prove the absurdity of an infinite succession, because it’s simply at the beginning of thought. They want to say something like a first intelligible.

The Debate Over the Concept “First Intelligible”

And here there is again politics between Maimonides and the Kalam, what is the meaning of first intelligible. It’s written here “at the beginning,” Ibn Tibbon translated “at the beginning of thought,” but I got confused here, because above we saw that “beginning of thought” is what Maimonides calls imagination, but they certainly aren’t claiming that they know this through imagination or through some unfounded opinion.

But apparently something, first of all this is not the same word in Arabic. Here it’s written literally in Arabic, something like this, at the beginning, not at the beginning of thought or the beginning of common knowledge as written above. These are all literal translations, that’s what’s written.

Aristotle’s First Intelligibles

So in other words, what’s happening here is that they’re claiming that this is a kind of, yes, we know, Aristotle had such a concept called first intelligible. First intelligible means the first intelligibles that are the foundations of all proof and we don’t need and cannot make a proof for them themselves, yes? Like that the whole is greater than the part, such things that we know, you can call this intuitively, but Aristotle calls this first intelligible.

Plato will say that this must be born with the soul, all kinds of such things, because there really is a problem of infinite regress, because of the principle that infinite causes cannot exist, so all thought must start somewhere, and Aristotle claimed that it starts from these first intelligibles, and these are things that don’t require proof, things that are self-evident, self-contradictory, anyone can look and see that this is true.

The Claim of the Mutakallimun

And now, apparently the Kalam, those of them who claimed that no proof is needed for the negation of this type of infinity, claim that this is of this type of things, this is a first intelligible, this is a beginning, this is itself a foundation. If you ask me how do I know that there cannot be something contradictory, because here knowledge begins, this is essentially the type of their answer.

The Misuse of First Intelligibles

But, Maimonides of course doesn’t think this is a first intelligible. And in general, this first intelligible is one of the things that is very abused, people use it wrongly, because they heard that Aristotle says there are things that don’t need proof, so anything that someone can, wants to imagine, then he starts to say, okay, but Aristotle said that no proof is needed to overturn, the most important things are the things that begin without proof, they are self-evident, as we tend to say today.

Al-Farabi’s Debate

And one of the arguments, the great debates of the Kalam and philosophy, in Al-Farabi’s book, which Maimonides already cited twice, On the Intellect and the Intelligible, he argues precisely on this point, that the Kalam don’t distinguish between what seems to me in a primary way, the beginning of thought or the beginning of common knowledge, and the true beginnings of knowledge, which are not anything that comes to your mind and is difficult for you to imagine, it is also a good beginning from which you can take an idea forward, like the first intelligible.

Maimonides’ Critique

So apparently this is what’s happening here, he writes about them a word that’s kind of moderate, because he writes that they say that this is self-evident, yes, this can be said to be self-proved, known at the beginning of thought, and that’s why Ibn Tibbon translated “beginning of thought,” because he really thinks like Maimonides that this is only the beginning of thought, and this is not a beginning in the sense of first intelligible.

But it’s not written here “beginning of thought,” they meant to say that this is a basic intelligible, contradictory, one can imagine, yes, Maimonides claims, yes, one cannot imagine this, yes, this goes, returns to the previous debate, but one can think it. Think about the difference between actual infinity and potential infinity, which are intelligible things, not imaginary, so this is essentially the debate here.

The Significance for the Question of the Eternity of the World

In any case, so this is the argument of the second type, they simply say that this is a first intelligible, no proofs are needed for the negation of this type of infinity. So now Maimonides concludes and says, and if this were true, if it were a clear falsehood, yes, falsehood, meaning negation, logical negation, clear, yes, that it’s clear, it’s proven, it’s a first intelligible or it has a proof, that there should be things without end in one respect of one after another, yes, in this sense, this aspect of infinity, of each one coming after the end of the previous thing, and even though each one of them is found to have an end, yes, not that there are infinite things, each one of them is finite, it has an end, but generally there is no end here, then it would be false, the eternity of the world from the beginning of thought.

The Decisive Implication

So the eternity of the world, it would be false within that same beginning of thought and would not require any other premise at all. If this were true then we would have paved over the eternity of the world at the beginning of the story. The whole great question at issue of the eternity or creation of the world wouldn’t begin because it would already be proven that the eternity of the world is simply a logically impossible statement. Yes?

And this is not the place for investigation of this matter, yes? And now we are not discussing the eternity of the world, we are only discussing the enumeration of the premises of the Kalam, but Maimonides just introduces us to the fact that essentially this premise is already the whole question, yes? The whole thing sought, yes, the fourth what… a quarter of the question, yes? The whole thing sought regarding the eternity of the world.

Summary: The True Innovation of the Mutakallimun

If they are really right in their understanding of the negation of the infinite, then the eternity of the world has already been proven, except that of course, as Maimonides explained, and for this he introduced the whole triple, quadruple, quintuple division of the infinite, to explain that their innovation is not that there is no infinity.

The Analogy to Accident

That there is no infinity in the most general way is true, similar to what he did regarding accident, yes? Everyone agrees that there is substance and accident, the question is what kind of substance and accident there is, yes?

The same thing, everyone agrees that actual infinity is impossible. The question is whether there is a distinction between actual infinity and potential infinity, the question is whether there is a distinction between infinity where all exist now and infinity that exist one after another, these are not questions, you may claim that this is a first intelligible or claim that you have a proof but these are not agreed-upon things and about this one really needs proof, needs to show.

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Okay, up to here is Premise 11, which is the premise on the negation of the infinite.

Thank you very much, listeners.