Revisiting the Argument from Motion

[inocente]

Good Grief Lucy! I never compared A. and T. A. with Christ - you are kidding, right?

Charlie

Emmm, it have not have been intentional, but the following is most definitely a comparison in anyone’s book:

many people reject A. and T, most of the world doesn’t accept Christ either. So should we stop talking about Christ because he is so unpopular?

Anyway, according to this 56% of the world accepts Jesus (Christian or Muslim), and the rest of the world doesn’t seem to mind, so I don’t see how you can claim He is so unpopular.

While A and TA’s fan base is tiny, and even then is perhaps outnumbered by opponents.

Besos y abrazos,

Lucy Van Pelt

 

[Linusthe2nd]

Infinite series do that to everyone I think I’ll get to Aquinas’ reformulations of Aristotle’s argument in terms of requiring a first term later as I would rather get the really complex argument out of the way first.

Aristotle’s argument which goes like this:

This argument has two flaws by my analysis. The reasoning itself is sound, premise 1 and 5 have problems, and I think that premise 10 requires clarification in regards to motion.

I’ve already given an important critique of premise in post 15 on page one. To elaborate:

Because causes often (indeed almost always) precede their effects rather than occurring simultaneously with them it is absurd to say that a mover and moved (which in Aristotelian terms is the same as saying an efficient cause and its effect) themselves always occur simultaneously. For example, If I set up a line of three dominoes and push the first one down, the first serves as the efficient cause of the fall of the second, which serves as the efficient cause of the fall of the third. And yet, the motion of the first, second, and third dominoes are not simultaneous.

One way to resolve this contradiction is to say that there is an intermediate causal agency of some kind. We would probably refer to this as simply a force in the case of physical motion. However, even so, it is clear from observation that there is no simultaneity in this case. If forces were also simultaneous then that would mean that domino three is simultaneously acted upon by the force from domino 2, domino 1, and the force imparted from my finger to domino 1 in the first place. This is clearly not the case as if it were so then the longer you make the domino chain the greater total force acting on the last domino would be as all of the previous forces would accumulate. As this clearly does not happen one must reject premise 1.

The other problem is premise 5. It is demonstrably not the case that an infinite series of positive values yields an infinite result. In mathematics an infinite series that adds up to a finite total is referred to as a convergent series. For example the infinite series 1+1/2+1/4+1/8… where each term is half the magnitude of the previous term the total result of this infinite series is two. Incidentally this is the basis of how Zeno’s dichotomy paradox is resolved in modern times. For most modern philosophers, Zeno actually got it right that motion involves passing over an actual infinity of locations. However, the total of time elapsed is not infinite as it is a convergent series rather than a divergent one.

This actually ties in with my issue with premise ten that an actual infinity cannot exist. I think that Zeno’s paradoxes are correct except in the case of concluding that infinite time required to complete an infinite series. Aristotle tries to escape the actual infinity of motion by saying that if you actually try to count out and divide the individual points you only have a potential infinity, but that’s missing the crux of the argument that the number of actual points themselves are infinite regardless of whether or not you discern them, and so it is an actual infinity.

I think a reasonable person of today would have to modify Aristotle’s assumption that actual infinities are impossible and instead say that only divergent actual infinities are impossible and that convergent actual infinities are not impossible.

Bakmoon, I really enjoy discussing things with you and I can’t say that about everyone here. You deserve a good answer which will take some time for reflection. The categories of Aristotle and Thomas Aquinas are difficult for the modern mind and require care to express correctly. But the problem here is the difference between per se and per accidens series. So in a day or several days I will get back to you.

Linus2nd

 

[Linusthe2nd]

Emmm, it have not have been intentional, but the following is most definitely a comparison in anyone’s book:

Anyway, according to this 56% of the world accepts Jesus (Christian or Muslim), and the rest of the world doesn’t seem to mind, so I don’t see how you can claim He is so unpopular.

While A and TA’s fan base is tiny, and even then is perhaps outnumbered by opponents.

Besos y abrazos,

Lucy Van Pelt

Lucy has her pig tails all in a knot! Why quibble over a few million, I haven’t bothered to count them. Any way I wasn’t comparing A. and TA to Jesus, you should know that. Give me a little credit for something.

Charlie

 

[Linusthe2nd]

Infinite series do that to everyone I think I’ll get to Aquinas’ reformulations of Aristotle’s argument in terms of requiring a first term later as I would rather get the really complex argument out of the way first.

Aristotle’s argument which goes like this:

This argument has two flaws by my analysis. The reasoning itself is sound, premise 1 and 5 have problems, and I think that premise 10 requires clarification in regards to motion.

I’ve already given an important critique of premise in post 15 on page one. To elaborate:

Because causes often (indeed almost always) precede their effects rather than occurring simultaneously with them it is absurd to say that a mover and moved (which in Aristotelian terms is the same as saying an efficient cause and its effect) themselves always occur simultaneously. For example, If I set up a line of three dominoes and push the first one down, the first serves as the efficient cause of the fall of the second, which serves as the efficient cause of the fall of the third. And yet, the motion of the first, second, and third dominoes are not simultaneous.

One way to resolve this contradiction is to say that there is an intermediate causal agency of some kind. We would probably refer to this as simply a force in the case of physical motion. However, even so, it is clear from observation that there is no simultaneity in this case. If forces were also simultaneous then that would mean that domino three is simultaneously acted upon by the force from domino 2, domino 1, and the force imparted from my finger to domino 1 in the first place. This is clearly not the case as if it were so then the longer you make the domino chain the greater total force acting on the last domino would be as all of the previous forces would accumulate. As this clearly does not happen one must reject premise 1.

The other problem is premise 5. It is demonstrably not the case that an infinite series of positive values yields an infinite result. In mathematics an infinite series that adds up to a finite total is referred to as a convergent series. For example the infinite series 1+1/2+1/4+1/8… where each term is half the magnitude of the previous term the total result of this infinite series is two. Incidentally this is the basis of how Zeno’s dichotomy paradox is resolved in modern times. For most modern philosophers, Zeno actually got it right that motion involves passing over an actual infinity of locations. However, the total of time elapsed is not infinite as it is a convergent series rather than a divergent one.

This actually ties in with my issue with premise ten that an actual infinity cannot exist. I think that Zeno’s paradoxes are correct except in the case of concluding that infinite time required to complete an infinite series. Aristotle tries to escape the actual infinity of motion by saying that if you actually try to count out and divide the individual points you only have a potential infinity, but that’s missing the crux of the argument that the number of actual points themselves are infinite regardless of whether or not you discern them, and so it is an actual infinity.

I think a reasonable person of today would have to modify Aristotle’s assumption that actual infinities are impossible and instead say that only divergent actual infinities are impossible and that convergent actual infinities are not impossible.

Here is a pretty good discussion on a per se or essential series verses a per accidens series. You might also read pgs 69-74 of Aquinas by Edward Feser.

In you comments above you have to be careful not to loose sight of the real " crux " of the whole issue, there is a real Unmoved Mover, God, behind every motion or change. It really does not matter if we satisfy everyone’s qualms about " infinities. " The point is that even if there are actual infinities ( which I do not admit ) they cannot exist on their own power, because, by the very nature of things, such infinities are composed of matter and form ( a principle of potentiality and a principle of actuality). And whether we run such things out to infinity or not they require a prime mover, a creator which is pure actuality and cannot, by that fact, be moved by anything else.

Linus2nd

 

[Linusthe2nd]

Infinite series do that to everyone I think I’ll get to Aquinas’ reformulations of Aristotle’s argument in terms of requiring a first term later as I would rather get the really complex argument out of the way first.

Aristotle’s argument which goes like this:

This argument has two flaws by my analysis. The reasoning itself is sound, premise 1 and 5 have problems, and I think that premise 10 requires clarification in regards to motion.

I’ve already given an important critique of premise in post 15 on page one. To elaborate:

Because causes often (indeed almost always) precede their effects rather than occurring simultaneously with them it is absurd to say that a mover and moved (which in Aristotelian terms is the same as saying an efficient cause and its effect) themselves always occur simultaneously. For example, If I set up a line of three dominoes and push the first one down, the first serves as the efficient cause of the fall of the second, which serves as the efficient cause of the fall of the third. And yet, the motion of the first, second, and third dominoes are not simultaneous.

One way to resolve this contradiction is to say that there is an intermediate causal agency of some kind. We would probably refer to this as simply a force in the case of physical motion. However, even so, it is clear from observation that there is no simultaneity in this case. If forces were also simultaneous then that would mean that domino three is simultaneously acted upon by the force from domino 2, domino 1, and the force imparted from my finger to domino 1 in the first place. This is clearly not the case as if it were so then the longer you make the domino chain the greater total force acting on the last domino would be as all of the previous forces would accumulate. As this clearly does not happen one must reject premise 1.

The other problem is premise 5. It is demonstrably not the case that an infinite series of positive values yields an infinite result. In mathematics an infinite series that adds up to a finite total is referred to as a convergent series. For example the infinite series 1+1/2+1/4+1/8… where each term is half the magnitude of the previous term the total result of this infinite series is two. Incidentally this is the basis of how Zeno’s dichotomy paradox is resolved in modern times. For most modern philosophers, Zeno actually got it right that motion involves passing over an actual infinity of locations. However, the total of time elapsed is not infinite as it is a convergent series rather than a divergent one.

This actually ties in with my issue with premise ten that an actual infinity cannot exist. I think that Zeno’s paradoxes are correct except in the case of concluding that infinite time required to complete an infinite series. Aristotle tries to escape the actual infinity of motion by saying that if you actually try to count out and divide the individual points you only have a potential infinity, but that’s missing the crux of the argument that the number of actual points themselves are infinite regardless of whether or not you discern them, and so it is an actual infinity.

I think a reasonable person of today would have to modify Aristotle’s assumption that actual infinities are impossible and instead say that only divergent actual infinities are impossible and that convergent actual infinities are not impossible.

Well, that is all over my head . That is why I said that the arguments 14, 15 in Book 1, ch 13 of the Summa Conrtra Gentiles were more meaningful to me - and conclusive…

Linus2nd

 

[Bakmoon]

Bakmoon, I really enjoy discussing things with you and I can’t say that about everyone here. You deserve a good answer which will take some time for reflection.

I also greatly enjoy this discussion. Philosophy is a great joy of mine and it is very rare to have a discussion on the internet that adheres even to the basic principles of rational discussion, to say nothing of philosophical subtlety. Being able to have a substantive conversation about Aristotle and Aquinas is a very rare thing and I am quite grateful for it.

Well, that is all over my head . That is why I said that the arguments 14, 15 in Book 1, ch 13 of the Summa Conrtra Gentiles were more meaningful to me - and conclusive…

Linus2nd

If you prefer Aquinas’ reformulation in the Summa Contra Gentiles how about we simply set aside Aristotle’s arguments and just focus on the argument from motion as formulated by Aquinas? I will convert his arguments into syllogystic form for analysis instead of Aristotle’s. Aquina’s version of it is much shorter so it’ll be much easier for both of us to analyse it I should think. I will do this for his first argument, and then after discussing the first argument we can go on to arguments two and three if that sounds good to you.

Here is a pretty good discussion on a per se or essential series verses a per accidens series. You might also read pgs 69-74 of Aquinas by Edward Feser.

On the distinction between a per se cause and a per accidens cause, I think there are points I could raise about how this distinction applies in the particular case of a proof for a first mover, but I think I will leave this discussion until after they come up in our discussion on Aquinas’ argument from motion.

 

[Linusthe2nd]

I also greatly enjoy this discussion. Philosophy is a great joy of mine and it is very rare to have a discussion on the internet that adheres even to the basic principles of rational discussion, to say nothing of philosophical subtlety. Being able to have a substantive conversation about Aristotle and Aquinas is a very rare thing and I am quite grateful for it.

If you prefer Aquinas’ reformulation in the Summa Contra Gentiles how about we simply set aside Aristotle’s arguments and just focus on the argument from motion as formulated by Aquinas? I will convert his arguments into syllogystic form for analysis instead of Aristotle’s. Aquina’s version of it is much shorter so it’ll be much easier for both of us to analyse it I should think. I will do this for his first argument, and then after discussing the first argument we can go on to arguments two and three if that sounds good to you.

On the distinction between a per se cause and a per accidens cause, I think there are points I could raise about how this distinction applies in the particular case of a proof for a first mover, but I think I will leave this discussion until after they come up in our discussion on Aquinas’ argument from motion.

That would be fine but we will have to limit ourselves to the First Way, form mothion, since that is the topic of the thread.

You give me way too much credit. Richa, Blato, Utunumsint are excellent.

Linus2bd

 

[Bakmoon]

That would be fine but we will have to limit ourselves to the First Way, form mothion, since that is the topic of the thread.

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.

You give me way too much credit. Richa, Blato, Utunumsint are excellent.

Linus2bd

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:

[1] We have now shown that the effort to demonstrate the existence of God is not a vain one. We shall therefore proceed to set forth the arguments by which both philosophers and Catholic teachers have proved that God exists.

[2] We shall first set forth the arguments by which Aristotle proceeds to prove that God exists. The aim of Aristotle is to do this in two ways, beginning with motion.

[3] Of these ways the first is as follows. Everything that is moved is moved by another. That some things are in motion—for example, the sun—is evident from sense. Therefore, it is moved by something else that moves it. This mover is itself either moved or not moved. If it is not, we have reached our conclusion—namely, that we must posit some unmoved mover. This we call God. If it is moved, it is moved by another mover. We must, consequently, either proceed to infinity, or we must arrive at some unmoved mover. Now, it is not possible to proceed to infinity. Hence, we must posit some prime unmoved mover.

[4] In this proof, there are two propositions that need to be proved, namely, that everything that is moved is moved by another, and that in movers and things moved one cannot proceed to infinity.

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.

[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].

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.

 

[utunumsint]

Hi Bakmoon,

Balto and I have just concluded our analysis of chapter 13 paragraph 5 through 10 on quid movetur ab alio movetur, and I think we made some good progress in developing our understanding of what Aquinas was saying. I’m ready to start delving into those relating to infinite regress. I am not personally ready to defend his claims yet. I still need to develop my understanding to a point where I can do this fruitfully (or concede appropriate counter arguments).

forums.catholic-questions.org/showthread.php?t=908045

If you have the time and the inclination, I would appreciate any help you can provide to enable us to develop our understanding. Perhaps after we have done the groundwork, we could work through your criticisms.

God bless,
Ut

 

[Bakmoon]

Hi Bakmoon,

Balto and I have just concluded our analysis of chapter 13 paragraph 5 through 10 on quid movetur ab alio movetur, and I think we made some good progress in developing our understanding of what Aquinas was saying. I’m ready to start delving into those relating to infinite regress. I am not personally ready to defend his claims yet. I still need to develop my understanding to a point where I can do this fruitfully (or concede appropriate counter arguments).

forums.catholic-questions.org/showthread.php?t=908045

If you have the time and the inclination, I would appreciate any help you can provide to enable us to develop our understanding. Perhaps after we have done the groundwork, we could work through your criticisms.

God bless,
Ut

I’d love to help but a lot of different things are being discussed in that thread. Is there something specific that I can lend my efforts to in that thread? I’m converting Aquinas’ first argument into a series of syllogisms for clarity’s sake, so how about once I finish that I go and post it over in the other thread to see if it makes Aquinas’ point clearer.

 

[utunumsint]

The other problem is premise 5. It is demonstrably not the case that an infinite series of positive values yields an infinite result. In mathematics an infinite series that adds up to a finite total is referred to as a convergent series. For example the infinite series 1+1/2+1/4+1/8… where each term is half the magnitude of the previous term the total result of this infinite series is two. Incidentally this is the basis of how Zeno’s dichotomy paradox is resolved in modern times. For most modern philosophers, Zeno actually got it right that motion involves passing over an actual infinity of locations. However, the total of time elapsed is not infinite as it is a convergent series rather than a divergent one.

This actually ties in with my issue with premise ten that an actual infinity cannot exist. I think that Zeno’s paradoxes are correct except in the case of concluding that infinite time required to complete an infinite series. Aristotle tries to escape the actual infinity of motion by saying that if you actually try to count out and divide the individual points you only have a potential infinity, but that’s missing the crux of the argument that the number of actual points themselves are infinite regardless of whether or not you discern them, and so it is an actual infinity.

I think a reasonable person of today would have to modify Aristotle’s assumption that actual infinities are impossible and instead say that only divergent actual infinities are impossible and that convergent actual infinities are not impossible.

I am not sure how one can pass through an infinity of points in a finite period of time. Can you unpack that for me or point me to resources that cover this idea of a convergent series?

The way I understand Zeno is that he was trying to argue for the impossibility of change. That change is an illusion. If we accept his paradox, then how can movement ever occur?

Also, the way I understand it, there are some physical limitations for the size of a given point. For example, Planck’s constant for a photon (a boson), as the smallest packet of energy possible in the physical universe. I know this is not a metaphysical proof, but it certainly shows that there is a limit to how far we can divide physical matter and still have entities that operate according to physical laws as we know them. Wouldn’t that make the idea of a convergent actual simply a logical possibility rather than something physically observable?

Perhaps if I had a better understanding of what you mean by a convergent actual, I could see the problem better.

God bless,
Ut

 

[Bakmoon]

I am not sure how one can pass through an infinity of points in a finite period of time. Can you unpack that for me or point me to resources that cover this idea of a convergent series?

Convergent infinite series are discussed in great detail the field of mathematics, especially in calculus, but you don’t necessarily need calculus to understand the concept.

In math a series is defined as a list of numbers that get added together. For example 1+2+3 is a series with three terms that happens to equal six. Series with a finite number of terms always add up to an actual number, and in the case of an infinite series, some of them add up to numbers and others don’t.

If an infinite series adds up to an actual number, this series is said to converge to that number, and is called a convergent series. If it doesn’t converge to a number but just keeps getting bigger and bigger without any kind of boundary, it is called a divergent series.

It is contrary to everyday intuition that an infinite series can converge to a number, but if you play around with the numbers you can see why it has to be the case. I know I already mentioned the example of the series 1+1/2+1/4… in which each term is half the previous term, but it is the traditional example usually given in math textbooks. If you calculate the first three terms you get 1.75. If you do the first 5 terms, you get 1.96875. If you put in the first 10 terms you get 1.99902. Using algebra you can find a formula for its partial sums, meaning if you put the number 5 into the formula, the answer tells you what the sum of the first 5 terms of the series is, etc… and when you find this formula and take the limit as it approaches infinity, you can prove that for the series 1+1/2+1/4… equals 2.

If you don’t like an algebra proof that an infinite series can add up to an actual number rather than becoming infinitely big there is a very nice geometrical proof for it. Consider the following diagram:

http://upload.wikimedia.org/wikipedia/commons/e/e6/GeometricSquares.svg

The first purple square on the bottom has the area 1/4, the one to the upper right of it has area 1/16, the one to the upper right of that one has area 1/64, etc so the total area of all the squares in purple is the series 1/4+1/14+1/64 etc… But all of these purple squares are contained within the big square that has an area of one, so the total area of the purple squares must be less than one. Because one is a finite number, the area of the purple squares must also be a finite number.

Even our system of decimal notation for certain fractions is based on the idea that certain infinite series converge. Every decimal we have can be rewritten as a series of fractions of powers of time. For example, 1.75 is the same as 1+7/10+5/100. But when we do this with repeating decimals we have an infinite series.

For example, if we convert 1 1/3 to decimal form we have 0.333… which is the same as the series 1+3/10+3/100+2/1000… so this series of fractions must add up to make 0.333 which is a finite number.

Convergent series are usually a shocking concept for most people when they first learn about them because they are counter intuitive, but that’s simply because we don’t actually work with infinities very often so of course our intuitions about it are likely off.

With the example of motion if you write a mathematical series that represents how long it takes to traverse an infinite series of points you find that although you have an infinite series, the series converges to an actual number, so there’s no problem.

 

[Bakmoon]

The way I understand Zeno is that he was trying to argue for the impossibility of change. That change is an illusion. If we accept his paradox, then how can movement ever occur?

We can accept Zeno’s reasoning as being valid without accepting all of his initial premises. If we do, then instead of being a disproof of motion, the argument becomes a reductio ad absurdum argument against the flawed premise. Zeno’s Dichotomy argument goes something like this:

1. Passing through an infinite number of submotions is an actual infinite
2. Motion involves passing through an infinite number of submotions.
3. Therefore, motion involves an actual infinite.
4. All actual infinities are impossible
5. Motion is an actual infinite
6. Therefore motion is impossible

Aristotle tries to refute the argument by saying that passing through an infinite number of points isn’t an actual infinite because it’s an infinity of division. I don’t think Aristotle’s argument works but I won’t get into that unless someone is interested. My point is that the Zeno’s deduction is sound, but one of his premises is unsound, and that is point 4, that all actual infinities are impossible. By making careful distinctions I think it makes sense to subdivide the category of actual infinities into subgroups, only some of which are impossible. For example, we can distinguish convergent and divergent series and say that the convergent series are allowed, thus motion is possible and the wacky infinities are still disallowed.

Also, the way I understand it, there are some physical limitations for the size of a given point. For example, Planck’s constant for a photon (a boson), as the smallest packet of energy possible in the physical universe. I know this is not a metaphysical proof, but it certainly shows that there is a limit to how far we can divide physical matter and still have entities that operate according to physical laws as we know them. Wouldn’t that make the idea of a convergent actual simply a logical possibility rather than something physically observable?

My mistake. I accidentally said in my post that Zeno’s argument is about an infinite number of points, but it’s actually an argument about an infinite number of submotions. Even so, I think the point variation idea also works.

Matter is not infinitely indivisible. That is correct. However, Zeno’s Dichotomy argument isn’t based on the infinite divisibility of matter but on an act of motion being divisible into an infinite number of submotions, or in the case of my accidental formulation, an infinite number of intermediate locations, and our current understanding of physics does allow an infinite subdivision of space.

 

[utunumsint]

We can accept Zeno’s reasoning as being valid without accepting all of his initial premises. If we do, then instead of being a disproof of motion, the argument becomes a reductio ad absurdum argument against the flawed premise. Zeno’s Dichotomy argument goes something like this:

1. Passing through an infinite number of submotions is an actual infinite
2. Motion involves passing through an infinite number of submotions.
3. Therefore, motion involves an actual infinite.
4. All actual infinities are impossible
5. Motion is an actual infinite
6. Therefore motion is impossible

Aristotle tries to refute the argument by saying that passing through an infinite number of points isn’t an actual infinite because it’s an infinity of division. I don’t think Aristotle’s argument works but I won’t get into that unless someone is interested. My point is that the Zeno’s deduction is sound, but one of his premises is unsound, and that is point 4, that all actual infinities are impossible. By making careful distinctions I think it makes sense to subdivide the category of actual infinities into subgroups, only some of which are impossible. For example, we can distinguish convergent and divergent series and say that the convergent series are allowed, thus motion is possible and the wacky infinities are still disallowed.

My mistake. I accidentally said in my post that Zeno’s argument is about an infinite number of points, but it’s actually an argument about an infinite number of submotions. Even so, I think the point variation idea also works.

Matter is not infinitely indivisible. That is correct. However, Zeno’s Dichotomy argument isn’t based on the infinite divisibility of matter but on an act of motion being divisible into an infinite number of submotions, or in the case of my accidental formulation, an infinite number of intermediate locations, and our current understanding of physics does allow an infinite subdivision of space.

Right. Thanks for clarifying. It is interesting that you bring this up, because I was just recently reading through a section in Robert J. Spitzer’s New Proofs for the Existence of God, where he states that…

"There must be certain finitely small natural minimums of space, time, and energy emissions within any universe, because these natural minimums prevent space and time from being reduced to infinitesimals (e.g., Euclidian points having position but no divisible magnitude).

No real continuum can be constituted by infinitesimals (which have no magnitude, interval, or “Delta”) because an infinite accumulation of such infinitesimals is still equal to zero magnitude. Now, if no real continuum can be constituted by infinitesimals, it must be constituted by some non-zero magnitude that is only finitely divisible. The point beyond which divisibility cannot proceed marks the finitely small (non-zero) natural minimum of space and time. One should therefore expect to find finitely small minimum units of space and time (where no further divisibility can really occur) in the universe. This is explained in greater detail in Chapter 5 (section II.C.5).

He then goes into the math which is above my head. Does any of this make sense to you? Is he saying something that contradicts you argument against point 5? He seems to be claiming that a convergent infinite series is only theoretically possible, but not physically possible based on what we know of our universe. I can delve into section II.C.5 if you need any further clarifications.

God bless,
Ut

 

[utunumsint]

Originally Posted by Bakmoon View Post
The other problem is premise 5. It is demonstrably not the case that an infinite series of positive values yields an infinite result. In mathematics an infinite series that adds up to a finite total is referred to as a convergent series. For example the infinite series 1+1/2+1/4+1/8… where each term is half the magnitude of the previous term the total result of this infinite series is two. Incidentally this is the basis of how Zeno’s dichotomy paradox is resolved in modern times. For most modern philosophers, Zeno actually got it right that motion involves passing over an actual infinity of locations. However, the total of time elapsed is not infinite as it is a convergent series rather than a divergent one.

This actually ties in with my issue with premise ten that an actual infinity cannot exist. I think that Zeno’s paradoxes are correct except in the case of concluding that infinite time required to complete an infinite series. Aristotle tries to escape the actual infinity of motion by saying that if you actually try to count out and divide the individual points you only have a potential infinity, but that’s missing the crux of the argument that the number of actual points themselves are infinite regardless of whether or not you discern them, and so it is an actual infinity.

I think a reasonable person of today would have to modify Aristotle’s assumption that actual infinities are impossible and instead say that only divergent actual infinities are impossible and that convergent actual infinities are not impossible.

Just to go back to your original argument - You seem to have moved from a position that a point is potentially divisible to infinity to a positive statement that it is an actual infinity, not merely a potential infinity. Is this warranted?

God bless,
Ut

 

[Bakmoon]

Right. Thanks for clarifying. It is interesting that you bring this up, because I was just recently reading through a section in Robert J. Spitzer’s New Proofs for the Existence of God, where he states that…

"There must be certain finitely small natural minimums of space, time, and energy emissions within any universe, because these natural minimums prevent space and time from being reduced to infinitesimals (e.g., Euclidian points having position but no divisible magnitude).

It’s an interesting quote but the author isn’t talking about convergent series. In that quote he is talking about something called an infinitesimal, which in mathematics is a non-real number that is infinitely small but is not equal to zero. Infinitesimals were first used as a way of working out calculus equations but they were extremely controversial when first proposed because it seems that infinitesimals don’t exist in the observable world. As a result calculus wasn’t universally accepted until a hundred years after it was invented when someone re-proved everything in calculus using limits instead of infinitesimals.

The author is arguing that infinitesimals don’t exist in the real world. I might disagree with some of his other conclusions regarding the subdivision of space, but I personally don’t have any problem with saying that infinitesimals are purely mathematical constructs with no real world analogue.

Convergent series however, are not made up of infinitesimals. If you look at the series 1+1/2+1/4… every single term in the series is a real number, meaning it actually has a position on the number line. Infinitesimals (and infinities as well, I might as well add) are most definately not real numbers because you can’t point to a spot on the number line and say “Here it is.” Every term in this convergent series is a real, finite, non-zero, non-infinitesimal number, and yet the sum of this infinite series of such numbers is itself an actual number.

In a little bit I’ll put my argument against point 5 in the form of a syllogism and post it. (If you haven’t noticed by now, my favorite method of analyzing arguments is to put them in syllogisms. Nothing beats syllogisms for the sake of clarity if you ask me!)

 

[utunumsint]

It’s an interesting quote but the author isn’t talking about convergent series. In that quote he is talking about something called an infinitesimal, which in mathematics is a non-real number that is infinitely small but is not equal to zero. Infinitesimals were first used as a way of working out calculus equations but they were extremely controversial when first proposed because it seems that infinitesimals don’t exist in the observable world. As a result calculus wasn’t universally accepted until a hundred years after it was invented when someone re-proved everything in calculus using limits instead of infinitesimals.

The author is arguing that infinitesimals don’t exist in the real world. I might disagree with some of his other conclusions regarding the subdivision of space, but I personally don’t have any problem with saying that infinitesimals are purely mathematical constructs with no real world analogue.

Convergent series however, are not made up of infinitesimals. If you look at the series 1+1/2+1/4… every single term in the series is a real number, meaning it actually has a position on the number line. Infinitesimals (and infinities as well, I might as well add) are most definately not real numbers because you can’t point to a spot on the number line and say “Here it is.” Every term in this convergent series is a real, finite, non-zero, non-infinitesimal number, and yet the sum of this infinite series of such numbers is itself an actual number.

In a little bit I’ll put my argument against point 5 in the form of a syllogism and post it. (If you haven’t noticed by now, my favorite method of analyzing arguments is to put them in syllogisms. Nothing beats syllogisms for the sake of clarity if you ask me!)

Cool. Thanks for clarifying. But would a convergent series be a Cantor set?

God bless,
Ut

 

[Bakmoon]

Just to go back to your original argument - You seem to have moved from a position that a point is potentially divisible to infinity to a positive statement that it is an actual infinity, not merely a potential infinity. Is this warranted?

God bless,
Ut

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:

Aristotle’s argument which goes like this:

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

The other problem is premise 5. It is demonstrably not the case that an infinite series of positive values yields an infinite result. In mathematics an infinite series that adds up to a finite total is referred to as a convergent series. For example the infinite series 1+1/2+1/4+1/8… where each term is half the magnitude of the previous term the total result of this infinite series is two. Incidentally this is the basis of how Zeno’s dichotomy paradox is resolved in modern times. For most modern philosophers, Zeno actually got it right that motion involves passing over an actual infinity of locations. However, the total of time elapsed is not infinite as it is a convergent series rather than a divergent one.

This actually ties in with my issue with premise ten that an actual infinity cannot exist. I think that Zeno’s paradoxes are correct except in the case of concluding that infinite time required to complete an infinite series. Aristotle tries to escape the actual infinity of motion by saying that if you actually try to count out and divide the individual points you only have a potential infinity, but that’s missing the crux of the argument that the number of actual points themselves are infinite regardless of whether or not you discern them, and so it is an actual infinity.

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

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. Passing through an infinite number of submotions is an actual infinite
2. Motion involves passing through an infinite number of submotions.
3. Therefore motion involves an actual infinite.

1. That which is directly observed is possible
2. Motion is directly observed
3. Therefore motion is possible

1. Let us assume for the sake of argument “If X is an actual infinite, then X is impossible”
2. Motion is an actual infinite
3. Therefore by our initial assumption, motion is impossible

1. If a statement leads to a contradiction with true statements, then such a statement is proven false by means of a reductio ad absurdum
2. The statement assumed in point 7 (i.e. “If X is an actual infinite, then X is impossible”) leads to a contradiction with a true statement (point 6)
3. Therefore the statement assumed in point 7 (“If X is an actual infinite, then X is impossible”) is proven false by means of a reductio ad absurdum

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

 

[Bakmoon]

Cool. Thanks for clarifying. But would a convergent series be a Cantor set?

God bless,
Ut

No. Every member of the cantor set is an infinitesimal point. None of the terms of a convergent series are infinitesimal (unless you really felt like making one up, and even then such a series would only work in some of the stranger alternate number systems).

 

[Bakmoon]

For example, if we convert 1 1/3 to decimal form we have 0.333… which is the same as the series 1+3/10+3/100+2/1000… so this series of fractions must add up to make 0.333 which is a finite number.

…

With the example of motion if you write a mathematical series that represents how long it takes to traverse an infinite series of points you find that although you have an infinite series, the series converges to an actual number, so there’s no problem.

Ack. More typos. The underlined fraction should read 3/1000 and the underlined word points should read submotion