Revisiting the Argument from Motion

[Bakmoon]

I finished turning Aquinas’ argument against an infinite series of movers into a series of syllogisms. In Book 13 of his Summa Contra Gentiles he states:

[11] The second proposition, namely, that there is no procession to infinity among movers and things moved, Aristotle proves in three ways.

[12] The first is as follows [VII, 1]. If among movers and things moved we proceed to infinity, all these infinite beings must be bodies. For whatever is moved is divisible and a body, as is proved in the Physics [VI, 4]. But every body that moves some thing moved is itself moved while moving it. Therefore, all these infinites are moved together while one of them is moved. But one of them, being finite, is moved in a finite time. Therefore, all those infinites are moved in a finite time. This, however, is impossible. It is, therefore, impossible that among movers and things moved one can proceed to infinity.

[13] Furthermore, that it is impossible for the abovementioned infinites to be moved in a finite time Aristotle proves as follows. The mover and the thing moved must exist simultaneously. This Aristotle proves by induction in the various species of motion. But bodies cannot be simultaneous except through continuity or contiguity. Now, since, as has been proved, all the aforementioned movers and. things moved are bodies, they must constitute by continuity or contiguity a sort of single mobile. In this way, one infinite is moved in a finite time. This is impossible, as is proved in the Physics [VII, 1].

Converting this into a series of syllogisms and filling in the implied terms I came to this:

1. All movers in an IRM (an infinite regress of movers) are bodies
2. A body which moves another body moves simultaneously and contiguously with it
3. By induction, all the movers in an IRM move simultaneously and contiguously

1. All movers in an IRM move simultaneously and contiguously
2. One mover will move a given amount in some finite amount of time
3. Therefore all the movers in such an IRM move simultaneously and contiguously in such a finite time.

1. The total motion of a series of movers (such as an IRM) is the sum of their individual motions
2. If a series has infinitely many members its total amount is infinite
3. Therefore the total motion of a series of infinite movers (such as an IRM) is infinite

1. An infinity of motion to occur in a finite time
2. An IRM has a infinity of motion in a finite time
3. Therefore an IRM is an actual infinite

1. An actual infinite is impossible
2. An IRM is an actual infinite
3. Therefore an IRM is impossible

I think this Aquinas’ argument expressed in its essential form. Does anyone suggest any modifications? If not, I’ll continue with my critique.

 

[Linusthe2nd]

Of Course. Detailed analysis of all of Aquinas’ arguments for the existence of God is far too wide a topic for a single forum thread.

I spend a lot of my time on this site just reading rather than posting, so I know the Philosophy forum has many very articulate and sensible philosophically minded individuals. But I don’t post often, so you are the first person with whom I’ve been able to have a direct and substantive philosophical discussion.

In Book I Chapter 13 of the Summa Contra Gentiles, St. Thomas Aquinas lays out his argument from motion like so:

Because my criticism of the argument from motion specifically links to the possibility of an infinite series of movers, for the purpose of this debate I am conceding Aquinas’ point that everything moved is moved by another and omitting his proof for this in points 5-10.

Aquinas gives three separate arguments against an infinite procession of efficient causes, but for now I will focus on his first argument and will move on to discuss his second and third after we have discussed the first.

It might take me a day or two to convert this argument into a series of syllogisms because Aquinas has more implied points in his argument and refers the reader back to the Physics, so I will have to explicitly state Aquinas’ implied points of argument and briefly summarize the relevant arguments of Aristotle which Aquinas cites, and I want to give a fair presentation of Aquinas’ argument without misrepresentation.

I won’t be commenting on that, since, as I said, I don’t really understand it. However the arguments Thomas gives in par 14,15 in book 1, ch 13 of SCG seem clear enough to make the case.

Linus2nd

 

[utunumsint]

I don’t think my original argument said anything about points. I think I mentioned them when describing Zeno’s Dichotomy argument but that’s just a memory lapse that should say submotion instead. In my objection to Aristotle’s argument from motion I said:

The two underlined portions are the places where I misstated Zeno’s argument. I should have used the word submotion.

Is there that much of a difference though? A motion presupposes continuous matter or space conjoined together where point A is conjoined to point B so that a thing can pass from A to B. And likewise the motion itself is of a physical substance that will also internally or externally move itself so that it can go from point A to point B.

My point is that just because a series is infinitely long doesn’t mean that the total of adding up all of them is itself infinity. Converting Zeno’s dichotomy argument against motion into a reductio ad absurdum argument against the impossibility of all actual infinities we get this argument:

1. If X moves Y then X and Y are simultaneous
2. In an infinite causal series ending in A, A is moved by B which is moved by C etc…
3. Therefore by induction, all the terms of such an infinite causal series occur simultaneously

1. The total amount motion for causal series is equal to the sum of the motion of its terms
2. An infinite sum of non-zero positive (like motion) amounts is infinite
3. Therefore an infinite causal series has an infinite amount of motion

1. An infinite amount of something occurring in the present is an actual infinity
2. An infinite causal series has an infinite amount of something (i.e. motion) as its terms occur simultaneously
3. An infinite causal series is an actual infinity

1. An actual infinity cannot exist
2. An infinite causal series is an actual infinity
3. Therefore an infinite causal series cannot exist

I think that this is a good argument against Aristotle’s unquallified statement that all actual infinities are impossible.

Right - Although I agree that you can logically divide a thing to infinity, just as you can calculate Pie to infinity, I don’t believe that in the physical world, this is actually possible. The quote I provided about the finite limits to space, time, and matter seems to corroborate this. That makes this discussion a issue of the underlying physics. If the Thomistic psychology is correct, then the only reason we can conceptualize an actual infinity is because, at least at this level of human cognition, we are engaging immaterial conceptual realities. Math is constantly prone to such idealizations and can become divorced from the physical reality.

I believe Liebnitz, and idealist, had some words to say about convergent series.

Also, just to clarify, your argument basically challenges the validity of your point 10, right? If 10 is wrong, the entire argument gets derailed. But I am not sure that your point about converget series would actually derail that statement. As you say, it would at most only refine that statement so that there would be at least one counter-example. Or am I missing something here.

God bless,
Ut

 

[Bakmoon]

Is there that much of a difference though? A motion presupposes continuous matter or space conjoined together where point A is conjoined to point B so that a thing can pass from A to B. And likewise the motion itself is of a physical substance that will also internally or externally move itself so that it can go from point A to point B.

It is a very significant difference. The continuity of space does not require the actual existence of infinitesimal points, but merely requires that it be possible to divide space into as many subintervals as you like.

Right - Although I agree that you can logically divide a thing to infinity, just as you can calculate Pie to infinity, I don’t believe that in the physical world, this is actually possible.

Do you mean to say that the infinite subintervals of a convergent series of motion or space in fact do not physically exist? If I consider the area of space between my bed and my door and subdivide it into an infinite number of subintervals forming a series that converges to the total distance, each one of these subintervals is an actual portion of the space between the bed and the door.

The quote I provided about the finite limits to space, time, and matter seems to corroborate this. That makes this discussion a issue of the underlying physics. If the Thomistic psychology is correct, then the only reason we can conceptualize an actual infinity is because, at least at this level of human cognition, we are engaging immaterial conceptual realities. Math is constantly prone to such idealizations and can become divorced from the physical reality.

I disagree with that books contention that time and space cannot be infinitely divided. The book argues that you can’t divide space and time down into *infinitesimals

• which probably has some merit to it, but that’s not the same thing as being able to divide space and time down to an infinite number of finite intervals which when added together form a convergent series.

I believe Liebnitz, and idealist, had some words to say about convergent series.

Indeed Liebnitz did. As the co-discoverer of calculus, he had to discover how to use convergent series to calculate definite integrals, so he certainly thought they existed and added up to real numbers otherwise he wouldn’t have thought that any of his work on integrals would have been valid proofs.

Also, just to clarify, your argument basically challenges the validity of your point 10, right? If 10 is wrong, the entire argument gets derailed. But I am not sure that your point about converget series would actually derail that statement. As you say, it would at most only refine that statement so that there would be at least one counter-example. Or am I missing something here.
God bless,
Ut

My converted form of Zeno’s dichotomy argument only provides a single counter example, true. However, it provides a counter example in terms of motion, and since motion is the subject matter of the argument from motion, it refutes point 10 in a way that applies directly to the argument at hand.

I have another objection in his treatment of simultaneity of movers. Aquinas rightly points out that a mover and the thing it moves undergo motion simultaneously. However, he assumes that this simultaneity is a transitive relation (meaning if A moves B and their movement is simultaneous, and B moves C and their action is simultaneous, we can conclude that A and C move simultaneously.) This is only true when they are part of the very same act of moving, however. For example, it is true that if I move a stick to push a rock, my moving the stick and the stick’s moving the rock are simultaneous. However, if I throw a stick which hits a rock and moves the rock, my initial act of throwing is not simultaneous with the moving of the rock, but the movement of the rock occurs after.

Since this is the case, if you have an infinite regress of movers being moved by other movers, the movers don’t have to be simultaneous, and therefore in such a case you don’t have an actual infinite but merely a potential infinite, which even by the standards of Aristotle and Aquinas, is not impossible.

 

[utunumsint]

It is a very significant difference. The continuity of space does not require the actual existence of infinitesimal points, but merely requires that it be possible to divide space into as many subintervals as you like.

Do you mean to say that the infinite subintervals of a convergent series of motion or space in fact do not physically exist? If I consider the area of space between my bed and my door and subdivide it into an infinite number of subintervals forming a series that converges to the total distance, each one of these subintervals is an actual portion of the space between the bed and the door.

Yes. That is what I am saying. I believe it is only potentially infinite in the same way that a line is potentially infinite or that you can move along the circumference of a circle for an potentially infinite time.

But I am no mathematician. Spitzer has a chapter about infinities where he references Cantor, but also David Hilbert. He wrote a chapter called “On the Infinite” in Philosophy of Mathematics. He explicitly rejects that any actual infinity is possible in the physical world, and that any actual infinity undermines the principles of finite math. Not being a mathematician, I have no way of knowing if he is right, but I can take your word for it, or his. As a non expert, I am sort of at the mercy of the experts. But, if you are interest, here is a quote:

A C-Infinity, like a B-infinity, is both a mathematical infinity and an applicationof infinity to a finite or aggregative structure. The important difference, however, between the B and C-infinities is that the C-infinity implies the existence (actuality) of a mathematical infinity within a finite or aggregative structure. As noted above, examples of C-infinities would be an actual Cantorian set with an actual infinite number of members, or an infinite number line with an actual infinite number of positions, or an actually existing infinite spatial magnitude, or an actual occurrence of an infinite number of events in the past. This, if C-infinities could really exist, there could be infinite space, infinity degrees Fahrenhiet, infinite mass density, infinite physical force, and infinite past time. These notions seem irresolvably pradoxical prima facie, because the mathematical infinity applied to them destroys their intelligibility as finite or aggregative structures. The proof for this goes beyong prima facie intuition. It extends to the requirements for mathematical intelligibility itself. Thus, as Hilbert shows, a mathematical infinity existing within a finite or aggregative structure undermines the very possibility of finite mathematics, and therefore the very possibility of quantifying those finite or aggregative structures. Therefore, a C-infinity must, in all cases, be illusory.

Or at least, Hilbert has convinced Spitzer that this is the case. As a mathematician (I presume that you are) you would probably have to look at Hilber’s work itself to see if you find it convincing or not. Assuming you don’t already know about it.

I disagree with that books contention that time and space cannot be infinitely divided. The book argues that you can’t divide space and time down into *infinitesimals

• which probably has some merit to it, but that’s not the same thing as being able to divide space and time down to an infinite number of finite intervals which when added together form a convergent series.

As per my quote above, I think Hilbert’s opinion covers any example of an actual infinity.

I’ll respond to the rest of your post later on today.

God bless,
Ut

 

[Bakmoon]

Yes. That is what I am saying. I believe it is only potentially infinite in the same way that a line is potentially infinite or that you can move along the circumference of a circle for an potentially infinite time.

Let’s clear up the terminology on actual and potential infinity.

In the case of an actual infinite, whatever thing is infinite is actually present rather than being merely potentially present, and the group is actually present as a whole within some period, whereas with a potential infinity, even if the individual parts of the whole are present actually, the infinity as a whole is never actually present within a particular period, and so it is called a potential infinity.

For example, suppose a computer starts counting from one and never stops. This is a potential infinite because although the total amount of numbers counted is infinite and they are actually present rather than being potentially present, it doesn’t occur in a finite amount of time, and so the existence of the whole thing is merely potential, so it is called a potential infinity.

However, if we try to imagine the computer counting an infinite amount of numbers in a finite time period, this is an actual infinity because all of the instances of counting are actually present within a single well defined period, so it is classified as an actual infinite.

In the case of Zeno’s dichotomy paradox every sub-interval of motion is present, not merely potentially, but actually, and the group of these infinitely many sub-intervals as a whole is actually present within a fixed series of time, so by definition it is an actual infinity, not a potential one, under Aristotelian definitions.

But I am no mathematician. Spitzer has a chapter about infinities where he references Cantor, but also David Hilbert. He wrote a chapter called “On the Infinite” in Philosophy of Mathematics. He explicitly rejects that any actual infinity is possible in the physical world, and that any actual infinity undermines the principles of finite math. Not being a mathematician, I have no way of knowing if he is right, but I can take your word for it, or his. As a non expert, I am sort of at the mercy of the experts. But, if you are interest, here is a quote:

Yes, I’ve read David Hilbert’s paper On the Infinite. All the paper argues is that there is no reason to conclude that actual infinities exist in nature. I don’t disagree that as a general principle actual infinities don’t exist in the physical world. I just think that there are specific exceptions, such as the case of convergent series.

One could of course, simply just redefine what we mean by potential and actual infinities in a way that moves the category of convergent series over into the category of potential infinities. I actually think that would probably be the best route to go. However, if you do that, then that means a convergent infinite regress of movers will be classified as a potential infinity which is possible, and thus the argument from motion will fail.

The way I see it, the argument from motion is stuck between a rock and a hard place on this issue. If you say that a convergent series of motion is an actual infinite, then you have to say that some actual infinite things are possible because because of my variation on Zeno’s dichotomy argument. If you say that a convergent series is merely a potential infinity, then an infinite regress of movers that is convergent is a potential infinite and thus is possible. The only way to make the argument from motion work is to show that a convergent series is potential, but a regress of movers is not.

 

[Bakmoon]

I’d like to reiterate a much larger criticism however, that even if a convergent infinite somehow is impossible, the argument from motion only works if you can show that there is an infinite chain of movers in a finite amount of time. If the chain of movers extends backwards into the past forever, then it is not an actual infinity. Both Aristotle and Aquinas know this, so they both make the argument that all of the movers in a chain of movers are simultaneous.

Like I have mentioned before though, although it is true that when a mover moves something the motion of both of them is simultaneous, this is only the case with two objects. If you have more than two objects it is possible for A to move B, and then for B to be set into motion and move C slightly later. In such a chain, not all of the movers are simultaneous. If we consider an infinite chain extending into the past, such a chain is only potentially infinite because not all the “links in the chain” are actually present within a finite time period.

 

[utunumsint]

Let’s clear up the terminology on actual and potential infinity.

In the case of an actual infinite, whatever thing is infinite is actually present rather than being merely potentially present, and the group is actually present as a whole within some period, whereas with a potential infinity, even if the individual parts of the whole are present actually, the infinity as a whole is never actually present within a particular period, and so it is called a potential infinity.

For example, suppose a computer starts counting from one and never stops. This is a potential infinite because although the total amount of numbers counted is infinite and they are actually present rather than being potentially present, it doesn’t occur in a finite amount of time, and so the existence of the whole thing is merely potential, so it is called a potential infinity.

However, if we try to imagine the computer counting an infinite amount of numbers in a finite time period, this is an actual infinity because all of the instances of counting are actually present within a single well defined period, so it is classified as an actual infinite.

In the case of Zeno’s dichotomy paradox every sub-interval of motion is present, not merely potentially, but actually, and the group of these infinitely many sub-intervals as a whole is actually present within a fixed series of time, so by definition it is an actual infinity, not a potential one, under Aristotelian definitions.

Agreed.

Yes, I’ve read David Hilbert’s paper On the Infinite. All the paper argues is that there is no reason to conclude that actual infinities exist in nature. I don’t disagree that as a general principle actual infinities don’t exist in the physical world. I just think that there are specific exceptions, such as the case of convergent series.

One could of course, simply just redefine what we mean by potential and actual infinities in a way that moves the category of convergent series over into the category of potential infinities. I actually think that would probably be the best route to go. However, if you do that, then that means a convergent infinite regress of movers will be classified as a potential infinity which is possible, and thus the argument from motion will fail.

The way I see it, the argument from motion is stuck between a rock and a hard place on this issue. If you say that a convergent series of motion is an actual infinite, then you have to say that some actual infinite things are possible because because of my variation on Zeno’s dichotomy argument. If you say that a convergent series is merely a potential infinity, then an infinite regress of movers that is convergent is a potential infinite and thus is possible. The only way to make the argument from motion work is to show that a convergent series is potential, but a regress of movers is not.

Right. You make a good point.

Well, here is what Hilbert says in a quote from Fr. Spitzer’s book.

Just as in the limit processes of the infinitesimal calculus, the infinite in the sense of the infinitely large and the infinitely small proved to be merely a figure of speech, so too we must realize that the infinite in the sense of an infinite totality, where we still find it used in deductive methods, is an illusion.

Now, because the convergent series you are proposing involves an infinity of smaller and smaller parts (and motions), I could use this quote (at least for myself) to justify rejecting it as either an instance of actual or potential infinity. Hilbert says a potential infinity is possible, but an actual infinity is impossible. If we redefine the convergent series as merely a potential series, well, it will still be complete, as you say, therefore actual. But it can’t be so, because of Hilbert’s prohibition against actual infinities, who must have known about convergent series, and must have rejected them as an exception as well.

Again, I am appealing to authority here, since I don’t have the skills necessary to personally prove or disprove my point. Perhaps if you could find a quote from Hilbert that deals with convergent series as bona fide exceptions to the rule, then my appeal to authority would fail.

If I couple Hilbert’s opinion with Fr. Spitzer’s point that space, time and matter cannot be infinity divided, then I feel like I have a pretty strong case in my support for the Aristotle’s dictum that an actual infinity is impossible.

I realize, of course, that you disagree with me on both points. I’m neither a mathematician or a physicist, so it would be hard for me to debate you on either point. But, there it is.

I still have not looked too deeply into the problem of simultaneous series yet. I’ll have to look into that before I can respond to your other point.

God bless,
Ut

 

[Bakmoon]

Now, because the convergent series you are proposing involves an infinity of smaller and smaller parts (and motions), I could use this quote (at least for myself) to justify rejecting it as either an instance of actual or potential infinity. Hilbert says a potential infinity is possible, but an actual infinity is impossible. If we redefine the convergent series as merely a potential series, well, it will still be complete, as you say, therefore actual. But it can’t be so, because of Hilbert’s prohibition against actual infinities, who must have known about convergent series, and must have rejected them as an exception as well.

I agree that convergent series are actual infinities by Aristotle’s definition. It’s a major part of my reformulation of the dichotomy argument. My modified dichotomy argument goes:

1. Let us assume that all actual infinities are impossible
2. A convergent series is an actual infinity.
3. Therefore a convergent series is impossible
4. An infinite series of finite parts which adds to a number is a convergent series
5. An act of motion can be subdivided into an infinite number of finite subintervals
6. Therefore an act of motion is a convergent series
7. A convergent series is impossible
8. Motion is a convergent series
9. Therefore motion is impossible

But everyone (except poor old Zeno) accepts that motion occurs and is possible, so we have a contradiction on our hands between saying that all actual infinities are impossible and saying that convergent series are actual infinities. The only way out of this contradiction is to redefine either our definition of an actual versus potential infinity to somehow fit convergent series into the category of potential infinity, or we can modify our position on actual infinities not being able to exist and say that only divergent infinities are impossible. If we don’t do one of these things, then one must conclude that motion is impossible.

Again, I am appealing to authority here, since I don’t have the skills necessary to personally prove or disprove my point. Perhaps if you could find a quote from Hilbert that deals with convergent series as bona fide exceptions to the rule, then my appeal to authority would fail.

Hilbert never addresses convergent series in his presentation. He gives many examples in his paper, but they are all examples of divergent infinities, so I think that the qualification is implied by context.

If I couple Hilbert’s opinion with Fr. Spitzer’s point that space, time and matter cannot be infinity divided, then I feel like I have a pretty strong case in my support for the Aristotle’s dictum that an actual infinity is impossible.

Hilbert never says that space and time cannot be infinitely divided, but only mentions matter and energy. And he doesn’t give a philosophical argument for this, but just points to quantum mechanics which says that energy is quantized.

I realize, of course, that you disagree with me on both points. I’m neither a mathematician or a physicist, so it would be hard for me to debate you on either point. But, there it is.

Hilbert’s presentation On The Infinite isn’t actually a technical math presentation. It’s a philosophical presentation that uses some examples from math, but I think that you could skip all of the math details and not miss very much. You can read the whole thing here: math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html

One thing to note though is that Hilbert doesn’t actually advance any kind of arguments in his presentation. It’s more of a general summary of the subject.

I still have not looked too deeply into the problem of simultaneous series yet. I’ll have to look into that before I can respond to your other point.

I look forward to your response then. Feel free to take your time to read up on the issue, as I’m not in any particular hurry to end this very fruitful discussion.

 

[utunumsint]

How you seen this article?

people.ucsc.edu/~jbowin/BOWA-2.1.pdf

Check out pages 10 and 11 - I think I will need to read it through a few times to absorb what the entire article is saying, but this seems to be talking about a convergent series, as you describe it, as a potential infinity.

God bless,
Ut

 

[utunumsint]

Really interesting:

It is complete and whole in so far as it is limited and contained by form, but only potentially so, because, as matter, form does not belong to it per se.

Wow - This sounds really close to what Aquinas says when he talks about essence and existence. No essence has existence per se except for God. Thus, because any physical object is such a composite of essence and existence, it cannot be an actual infinite.

God bless,
Ut

 

[Bakmoon]

How you seen this article?

people.ucsc.edu/~jbowin/BOWA-2.1.pdf

Check out pages 10 and 11 - I think I will need to read it through a few times to absorb what the entire article is saying, but this seems to be talking about a convergent series, as you describe it, as a potential infinity.

God bless,
Ut

Yes, I’ve seen that article. Aristotle himself said something similar in the case of Zeno’s dichotomy argument (what we would call a convergent series) in Book 6 Chapter 2 of the Physics where he says:

For there are two ways in which length and time and generally anything continuous are called infinite: they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility; for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.

Although Aristotle doesn’t to my knowledge explicitly classify an infinity of division as a type of potential infinity, I think context demands that we understand this to be the case because there are places in the Physics where Aristotle says that only potential infinities can exist just after talking about the addition-division distinction with infinities, so it wouldn’t make sense to say that Aristotle classifies an infinity of division as an actual infinity.

Looking at the contexts in which Aristotle actually talks about the infinity of division versus an infinity of addition in my opinion is quite illuminating on how Aristotle understood an actual infinity. In Book 6 Chapter 6 of the Physics he says something very important:

Now things are said to exist both potentially and in fulfilment. Further, a thing is infinite either by addition or by division. Now, as we have seen, magnitude is not actually infinite. But by division it is infinite. (There is no difficulty in refuting the theory of indivisible lines.) The alternative then remains that the infinite has a potential existence.

The key word here that I think most people miss is the word magnitude. In my reading of him, for Aristotle, it isn’t the infinite number of something that makes it impossible, but the idea of an infinite magnitude such as mass, temperature, etc… that makes it impossible. With Zeno’s dichotomy, although the number of subintervals is indeed infinite, you don’t have any physical magnitude being infinite, so Aristotle doesn’t have a problem with it.

This being the case I think one must clarify the absolute ban on actual infinities as referring only to infinities of physical magnitudes. Or at least, that’s how I would amend it. Or equivalently, you could amend the definition of an actual infinity to mean an infinity of magnitude, both work. In either case though, convergent series are allowed because their magnitude is finite.

 

[utunumsint]

Yes, I’ve seen that article. Aristotle himself said something similar in the case of Zeno’s dichotomy argument (what we would call a convergent series) in Book 6 Chapter 2 of the Physics where he says:

Although Aristotle doesn’t to my knowledge explicitly classify an infinity of division as a type of potential infinity, I think context demands that we understand this to be the case because there are places in the Physics where Aristotle says that only potential infinities can exist just after talking about the addition-division distinction with infinities, so it wouldn’t make sense to say that Aristotle classifies an infinity of division as an actual infinity.

Looking at the contexts in which Aristotle actually talks about the infinity of division versus an infinity of addition in my opinion is quite illuminating on how Aristotle understood an actual infinity. In Book 6 Chapter 6 of the Physics he says something very important:

The key word here that I think most people miss is the word magnitude. In my reading of him, for Aristotle, it isn’t the infinite number of something that makes it impossible, but the idea of an infinite magnitude such as mass, temperature, etc… that makes it impossible. With Zeno’s dichotomy, although the number of subintervals is indeed infinite, you don’t have any physical magnitude being infinite, so Aristotle doesn’t have a problem with it.

This being the case I think one must clarify the absolute ban on actual infinities as referring only to infinities of physical magnitudes. Or at least, that’s how I would amend it. Or equivalently, you could amend the definition of an actual infinity to mean an infinity of magnitude, both work. In either case though, convergent series are allowed because their magnitude is finite.

Humm - I check’s Aquinas’ commentary

dhspriory.org/thomas/Physics6.htm#6

He says this here:

However, it does not follow that since there are an infinitude of intermediates, the ultimate cannot be reached, because these intermediate places are infinite not in act but only in potency, just as a magnitude is not actually divided infinitely but is potentially divisible.

God bless,
Ut

 

[Frater_Bovious]

Hi all. Just a couple of quick comments.

1. 
   *]The Summa presents summaries of the five proofs. They presuppose his readers have an educational foundation in what we would call today Philosophy, which would have been taught back then prior to engaging in a study of Theology.
   *]Because of this, many people find fault with what they think the argument is, and there is also a tendency to gloss over points or principals that are actually pivotal.
   *]In this case, I think the concepts of potentiality and actuality are not fully appreciated. Since the argument depends on these concepts, the most efficient way to disprove the argument would be to disprove the underlying concepts.
   *]This thread is about the argument from motion (the first way), not the argument from efficient cause (the second way).
   *]Another issue is that when Aquinas says motion, he means what we today call change. His argument could be crudely summarized as “things don’t change by themselves.”
   
   So, I have to disagree with this:
   

   This is actually another problem I have with his argument as the whole argument assumes that causes and effects occur simultaneously.

   The only place that I can see where you could draw the conclusion that his whole argument assumes causes and effects occur simultaneously is with regard to Aquinas’s example of a hand pushing a stick. This could be seen as an example of contingent cause and effect, but this is the argument from motion, as noted above, and not from efficient causes, and so maybe the two arguments are conflated in your mind? They are similar, but the argument from motion does not require simultaneous cause and effect.
   
   Remembering that by “move” we can more profitably think “change”, anything that moves from potentiality to act (changes from just sitting there to bouncing, for example) has to be put into act by something else. A ball has the potential to bounce, but it does not bounce itself.
   
   To put it another way, if a basketball suddenly appeared on your kitchen table during dinner, you would not simply go, “Oh, interesting, a ball has manifested itself from nothing.” Let’s face it, you would say, “Where did that come from? How did it get here?”, betraying a belief that something moved it there.
   
   Another way people misconstrue this argument is by pointing out that, for example, dogs move themselves. I think that depends on your point of view, but I will simply concede that for the moment. It doesn’t change the fact that balls don’t bounce themselves.
   
   Things don’t just come into existence out of nothing by themselves either. Again, people will point to the creation and obliteration of particles with their anti-particles in the vacuum of space - and I just have to say that the vacuum energy of space is not “nothing” and something is causing those particles to change from potential to actual and back.
   
   Don’t know if any of that helps, but there you go.
   
   Mark Connolly, M.T.S.

 

[Bakmoon]

The Summa presents summaries of the five proofs. They presuppose his readers have an educational foundation in what we would call today Philosophy, which would have been taught back then prior to engaging in a study of Theology.

That’s very true of the Quinquae Viae as presented in the Summa Theologica. That’s why most of the discussion on the argument from motion has so far focused on it as presented in Book 7 of Aristotle’s Physics and the relevant portions from the Summa Contra Gentiles chapter 13, which give more extensive explanation.

In this case, I think the concepts of potentiality and actuality are not fully appreciated. Since the argument depends on these concepts, the most efficient way to disprove the argument would be to disprove the underlying concepts.

I personally think that disproving the actuality-potentiality distinction wouldn’t be sufficient to refute the argument from motion as the terms of the argument itself don’t explicitly draw on the subjects, so any theory of causation that doesn’t do violence to the ideas of mover and moved wouldn’t be a good enough refutation.

The only place that I can see where you could draw the conclusion that his whole argument assumes causes and effects occur simultaneously is with regard to Aquinas’s example of a hand pushing a stick. This could be seen as an example of contingent cause and effect, but this is the argument from motion, as noted above, and not from efficient causes, and so maybe the two arguments are conflated in your mind? They are similar, but the argument from motion does not require simultaneous cause and effect.

In the Summa Contra Gentiles Aquinas gives three arguments against the possibility of an infinite regress of movers, the first of which requires simultaneity. In Book 1 Chapter 13 Aquinas says:

[11] The second proposition, namely, that there is no procession to infinity among movers and things moved, Aristotle proves in three ways.

[12] The first is as follows [VII, 1]. If among movers and things moved we proceed to infinity, all these infinite beings must be bodies. For whatever is moved is divisible and a body, as is proved in the Physics [VI, 4]. But every body that moves some thing moved is itself moved while moving it. Therefore, all these infinites are moved together while one of them is moved. But one of them, being finite, is moved in a finite time. Therefore, all those infinites are moved in a finite time. This, however, is impossible. It is, therefore, impossible that among movers and things moved one can proceed to infini

Aquinas is saying that the entire chain of movers must take place together in a finite amount of time. I think the English translation here isn’t quite clear enough. The passage that goes “But every body that moves some thing moved is itself moved while moving it. Therefore, all these infinites are moved together while one of them is moved.” uses the word ‘while’ which isn’t clear enough in my opinion.

The original Latin reads: “Omne autem corpus quod movet motum, simul dum movet movetur. Ergo omnia ista infinita simul moventur dum unum eorum movetur.” The Latin adverb simul in this context has the meaning of ‘at the same time’, and that’s essential to Aquinas’ argument because he follows this statement by saying " Therefore, all these infinites are moved together while one of them is moved. But one of them, being finite, is moved in a finite time." and if Aquinas isn’t talking about simultaneous movers then he couldn’t conclude that the entire chain of movers must occur in a finite period of time on the basis of one of the movers moving in a finite period of time.

To put it another way, if a basketball suddenly appeared on your kitchen table during dinner, you would not simply go, “Oh, interesting, a ball has manifested itself from nothing.” Let’s face it, you would say, “Where did that come from? How did it get here?”, betraying a belief that something moved it there.

I have no problem in saying that (excluding quantum mechanical weirdness I don’t feel like geting into) changes require causes. My problem is saying that an infinite chain of movers must automatically be actually infinite and therefore impossible.

 

[Frater_Bovious]

Thanks for providing the other sources - I need to learn to stop jumping in the middle of an ongoing conversation without reading the thread - I have not read any of the Summa Contra Gentiles - I’ve been looking for a copy (I don’t like reading online) and the only place I’ve found copies is in the library of a local Dominican Priory. Can you recommend a translation?

Thanks.

Mark

 

[utunumsint]

Humm - I check’s Aquinas’ commentary

dhspriory.org/thomas/Physics6.htm#6

He says this here:

However, it does not follow that since there are an infinitude of intermediates, the ultimate cannot be reached, because these intermediate places are infinite not in act but only in potency, just as a magnitude is not actually divided infinitely but is potentially divisible.

Interesting. He also says this:

Likewise, it is not against the nature of magnitude that. every magnitude be divisible into smaller ones. Therefore, in this Book he goes on the assumption that it is possible to take a magnitude smaller than any given magnitude, even though in every particular nature there is always a minimum magnitude, since each nature has limits of largeness and smallness, as was mentioned even in Book I.

I wonder if what he is talking about here is prime matter’s ability to take on any form, or to move from one form to another. Basically, prime matter is infinite potentiality to be something else, to move from one form to another. Given this background metaphysics, perhaps the reason why he considers a convergent series as only a potential infinity, is on account of the fact that any thing that is a convergent series can change its form? And if it can change its form, then it is not an actual eternity in terms of time, but only in terms of place, for a finite time.

This would also fit into his natural theology, such that only a being that is not a composite of form and matter, but whose essence is existence itself, who is pure actuality, could possibly be eternal, strictly speaking. All other essences only have being in participation, and so are in potency to the divine. Aquinas also says that common being (ens commune) is the first effect caused by God. This common being is to be distinguished from God’s actus purus in the sense that God’s actus purus is pure actuality, while ens commune is almost pure potentiality to all form, genus, species.

What do you think?

God bless,
Ut

 

[utunumsint]

1. In regard to the first he lists four reasons, and this is what he says: Zeno used against motion four arguments which have caused difficulty for many of those who tried to answer them. The first of which is this: If anything is being moved through a certain space, it must reach the middle before it reaches the end. But since the first half is divisible, half of it must be first traversed and so on indefinitely, since a magnitude can be divided ad infinitum. Infinites, however, cannot be traversed in finite time. Therefore, nothing can be moved.

Therefore Aristotle says that he has already answered this argument (in the beginning of this Book VI), when he proved that time is divided ad infinitum in the same way as a magnitude is. This answer is directed more to one who asks whether infinites can be traversed in finite time than to the question, as he will say in Book VIII, (L. 17) where he answers this argument by showing that a mobile does not use the infinites which exist in a magnitude as though they were actually existing, but only as existing potentially. For a thing in motion uses a point in space as actually existing, when it uses it as a beginning and as an end, and it is then that the mobile must be at rest, as will be explained in Book VIII. **But if it had to traverse infinites that were actually existing, then it would never reach the end. **

If by actuality, one means form, as Aquinas does then, as per my previous post, any given thing does have actual limits. I also think that this is what Spitzer is getting at when he insists that there are real limits to space, time, and matter based on the Planck constant. There is a point, in terms of divisibility, by which there is no longer any discernable forms possible.

God bless,
Ut

 

[Bakmoon]

Thanks for providing the other sources - I need to learn to stop jumping in the middle of an ongoing conversation without reading the thread - I have not read any of the Summa Contra Gentiles - I’ve been looking for a copy (I don’t like reading online) and the only place I’ve found copies is in the library of a local Dominican Priory. Can you recommend a translation?

Thanks.

Mark

I personally do most of my reading from PDFs I get online, including the Summa Contra Gentiles so unfortunately I can’t make any recommendations for print books.

I used to hate using PDFs but then I switched over to using a PDF reader called PDF Xchange Viewer and it has bookmark options and such that make it quite convenient. It’s available for free, so you might consider trying it out.

 

[Bakmoon]

I wonder if what he is talking about here is prime matter’s ability to take on any form, or to move from one form to another. Basically, prime matter is infinite potentiality to be something else, to move from one form to another. Given this background metaphysics, perhaps the reason why he considers a convergent series as only a potential infinity, is on account of the fact that any thing that is a convergent series can change its form? And if it can change its form, then it is not an actual eternity in terms of time, but only in terms of place, for a finite time.

That’s a very good point. Most people forget that in Aristotelianism unorganized matter has an infinite potential. But I don’t think that’s quite how Aristotle would explain the possibility of a convergent infinite series. Aristotle does talk about convergent infinite series of motion (of course the actual terminology here is modern) in his discussion of Zeno’s dichotomy argument in Book 7 section 2 of the Physics, and he answers it by referring to the distinction between an infinity of divisibility and an infinity of extremity (which seems to be the equivalent of Book 3’s infinity of addition), and then by equating an infinity of extremity with an infinite magnitude. He says:

Hence Zeno’s argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time. For there are two ways in which length and time and generally anything continuous are called infinite: they are called so either in respect of divisibility or in respect of their extremities. So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility; for in this sense the time itself is also infinite: and so we find that the time occupied by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.

In other places too Aristotle frames the distinction between an infinity of addition and an infinity of division in terms of an infinity of magnitude. For example, in Book 3 Chapter 4 Aristotle says:

The problem, however, which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite…everything that is infinite may be so in respect of addition or division or both.

And in Book 3 Chapter 6 Aristotle similarly says:

Now things are said to exist both potentially and in fulfilment. Further, a thing is infinite either by addition or by division. Now, as we have seen, magnitude is not actually infinite. But by division it is infinite. (There is no difficulty in refuting the theory of indivisible lines.) The alternative then remains that the infinite has a potential existence.

I think that reading Aristotle’s discussions of infinity in context indicates that Aristotle is only concerned about the actual infinity of observable magnitudes.