Revisiting the Argument from Motion

[Bakmoon]

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

I disagree very strongly with Spitzer’s argument that space and time are not continuous. He is correct that matter and energy are not continuous as per our current understanding of quantum mechanics, but I haven’t seen anything convincing that indicates that space and time aren’t continuous. Space and time are described by the Einstein’s theory of General Relativity, and General Relativity has no problem with space and time being infinitely divisible. In fact, when General Relativity talks about space and time together (in General relativity they both governed by the same set of equations) they are usually referred to as the ‘Spacetime Continuum’

 

[utunumsint]

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:

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:

And in Book 3 Chapter 6 Aristotle similarly says:

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

Well then it seems that Aquinas is going beyond what Aristotle would allow then in stating that an infinity of division is only potential.

God bless,
Ut

 

[Bakmoon]

Well then it seems that Aquinas is going beyond what Aristotle would allow then in stating that an infinity of division is only potential.

God bless,
Ut

Not nessisarily. You could understand Aristotle as saying that only magnitudes count as actual in which case Aquinas’ understanding of an infinity of division as not being an actual infinity makes perfect sense.

 

[utunumsint]

Not nessisarily. You could understand Aristotle as saying that only magnitudes count as actual in which case Aquinas’ understanding of an infinity of division as not being an actual infinity makes perfect sense.

Well, I think both Aristotle and Aquinas would define any material actuality as stemming from the substantial form. Any division that destroys the intelligibility of the substantial form, and therefore the qualities, such as magnitude, that flow from it, would necessarily seem to disqualify that thing as an actual. At least in their terms. But again, that goes back to my point about prime matter never existing on its own without form.

God bless,
Ut

 

[Bakmoon]

Well, I think both Aristotle and Aquinas would define any material actuality as stemming from the substantial form. Any division that destroys the intelligibility of the substantial form, and therefore the qualities, such as magnitude, that flow from it, would necessarily seem to disqualify that thing as an actual. At least in their terms. But again, that goes back to my point about prime matter never existing on its own without form.

God bless,
Ut

When I said magnitude I was specifically referring to magnitude as opposed to mere cardinality (meaning the answer you get from counting something). My point was that for Aristotle, it seems that a convergent series isn’t an actual infinity because even though the number of subintervals is infinite, the number of subintervals is itself not a kind of magnitude. That is not to say that substantial form doesn’t have a role to play in the actual-potential distinction as well, but it doesn’t directly have relevance in terms of the criterion with which we decide between an actual and potential infinity.

 

[Arrowsmith1994]

I don’t think Aristotle was a polytheist. Socrates denied the existence of the Greek gods, and this is recorded by Plato who likely agreed with him. Can anyone provide evidence that Aristotle believed in the Greek gods and if not, what deities are you talking about?

 

[Jaaanosik]

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:

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:

And in Book 3 Chapter 6 Aristotle similarly says:

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

A convergent infinite series is a mathematical concept. It’s only good as far as it relates and describes the reality properly.
The actual infinity does not exist because of energy quanta. Motion and energy are inseparable therefore the motion happens only in quanta as well. The motion is not infinitely divisible. That’s our understanding of physics.

Philosophy, metaphysics, physics, mathematics, … it’s all about the truth. The truth is only one, these subjects can not contradict each other if they describe the same truth.

 

[Bakmoon]

A convergent infinite series is a mathematical concept. It’s only good as far as it relates and describes the reality properly.
The actual infinity does not exist because of energy quanta. Motion and energy are inseparable therefore the motion happens only in quanta as well. The motion is not infinitely divisible. That’s our understanding of physics.

Philosophy, metaphysics, physics, mathematics, … it’s all about the truth. The truth is only one, these subjects can not contradict each other if they describe the same truth.

It simply does not logically follow that if energy is quantized then motion or space must be simply because energy and motion are related. The actual energy involved in a motion must be quantized but that doesn’t mean that somehow the actual motion itself is discontinuous.

 

[utunumsint]

Hi Bakmoon,

I just wanted to summarize my case against actual infinities:

1. 
   *]I am still not convinced that Hilbert’s prohibition was not global. A mathematician of his caliber should not have made a global statement like that unless he believed there were no exceptions.
   *]The way I understand Aquinas is that infinite division is only a potential infinity. He does admit that this divisibility makes for a continuum, but not an actual infinity.
   *]Spitzer’s conclusions about quanta makes sense to me. I have read somewhere that motion is actually jumpy at the quantum level. Especially with regard to electron jumps from orbital to orbital. I’m not sure about how continuous or discontinuous the motion is between photon emission from atom to atom, but there it is. Things get strange at the quantum level.
   
   Either way, mathematics is deductive, and when it comes to reality, we need to empirically and inductively confirm deductive conclusions as much as possible. I don’t see how it is possible to empirically or inductively calculate an actual infinity, which by definition is not quantifiable. There may be a sense in which this question is fundamentally unanswerable in this way. However, this conclusion does fit in with Hilbert’s prohibition though, which is that actual infinites in the real world violate the principles of finite math.
   
   God bless,
   Ut

 

[utunumsint]

Just to go back to the John Bowin article,

In the discussion of place in Physics 4. 2 (209B5–8) Aristotle identifies the spatial extension of a magnitude with its matter, and the magnitude itself with a form/matter composite consisting of a bounding surface and a spatial extension respectively. If this is the definition of a magnitude, and the infinite ‘is the matter of the completeness which belongs to magnitude’, then the point of the passages just quoted appears to be that the infinite, understood as what is infinite, is to be identified with the material element or material cause of a form/matter composite. As matter, what is infinite is potentially but not actually complete and whole. 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.

So matter is what is considered infinite here. If matter can always be added to or divided, then it is completely unbounded. But when conjoined with form, it can be bounded.

Thus, what is infinite is unbounded, or :;$<6=0, in the sense that it is unlimited by anything intrinsic to it. The first passage also tells us that what is infinite, i.e. matter, is ‘divisible both in the direction of reduction and of the inverse addition’ (207A22–3), which I take to mean infinite divisibility. Aristotle has told us at 206B16–20 that divisibility in the direction of reduction, or the infinite in respect of division, is ‘in a sense the same’ as divisibility in the direction of the inverse addition, or the infinite in respect of addition. This amounts to the claim that the division of a magnitude according to an infinite geometric sequence such as …1/4, 1/8, [etc…] according to an infinite series… or rather according to a sequence of partial sums.The infinite divisibility of a magnitude by reduction and addition relates to the potential completeness and wholeness of
a magnitude in respect of the status of thematerial parts produced and added in this way. A magnitude is complete and whole if it has all of its material parts. But a magnitude has all of its material parts only potentially since new material parts may always be produced by additional divisions and added to the collection ofmaterial parts
already produced.

Interesting. Here he defines a magnitude (flowing from a form/matter composite) as only being potentially complete, because its matter can always be further divided, thus creating new parts

Thus, the infinite divisibility of a magnitude is the cause of its being complete and whole only potentially, and the material element, in turn, is the source of this infinite divisibility. Thus, potential infinity is a property that the material element of a form/matter composite (i.e. ‘the infinite’ understood as ‘what is infinite’) contributes to the form/matter composite, viz. its infinite divisibility, or conversely, its being filled out by a potentially infinite number of material parts.

Very interesting. So here, Aristotle is defining what you would call a convergent series, the very cause of a substance only having completeness potentially and not actually. Whereas you seem to define convergent series as being a reason to consider it an actual infinity…

This is bewildering.

God bless,
Ut

 

[Bakmoon]

I am still not convinced that Hilbert’s prohibition was not global. A mathematician of his caliber should not have made a global statement like that unless he believed there were no exceptions.

Or he could have simply thought that convergent series are potential infinities just as Aristotle thought.

*]The way I understand Aquinas is that infinite division is only a potential infinity. He does admit that this divisibility makes for a continuum, but not an actual infinity.

It is true that both Aristotle and Aquinas treat infinities of division as potential infinities. I have no problem with saying that. My point is that an infinity of division can have all of its elements present within a finite time as long as you don’t end up with infinite magnitudes. Aristotle makes this exact same point in his refutation of Zeno’s dichotomy argument in Book VI Section II of the physics where he says:

… 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.

Note that Aristotle agrees with Zeno that a moving thing does in fact come into contact with infinite things (i.e. subintervals of motion) in a finite time. For Aristotle it is still counted as an infinity of division because it doesn’t come into contact with anything that is quantitatively infinite. As a result, for Aristotle, this is an infinity of division, which he seems to classify as a potential infinity.

I have no problem with any of this. But my main point is this: By Aristotle’s standards as laid out in his discussion of the dichotomy argument, a convergent series of movers would also be classified as an infinity of division, and therefore a potential infinity, which is therefore possible. In both Aristotle’s and Aquinas’ arguments from motion, they never address this possibility, and so there is a hole in the argument.

Spitzer’s conclusions about quanta makes sense to me. I have read somewhere that motion is actually jumpy at the quantum level. Especially with regard to electron jumps from orbital to orbital. I’m not sure about how continuous or discontinuous the motion is between photon emission from atom to atom, but there it is. Things get strange at the quantum level.

It’s absolutely true that at the quantum level that some forms of motion are discontinuous. Quantum tunneling is a famous example in which a particle moves past a barrier that it cannot move through continuously just by vanishing and appearing on the other side.

However, this is only some kinds of motion, not all. On the quantum level ordinary motions can very well be continuous as well. Take for example the motion of a photon. A photon isn’t like a little billiard ball made out of light, but is a packet of an electro-magnetic wave that propagates throughout space continuously at the speed of light.

These kinds of properties aren’t just true for photons, though. All particles have a wave-like aspect to them, and these waves propagate continuously throughout space.

 

[utunumsint]

Or he could have simply thought that convergent series are potential infinities just as Aristotle thought.

It is true that both Aristotle and Aquinas treat infinities of division as potential infinities. I have no problem with saying that. My point is that an infinity of division can have all of its elements present within a finite time as long as you don’t end up with infinite magnitudes. Aristotle makes this exact same point in his refutation of Zeno’s dichotomy argument in Book VI Section II of the physics where he says:

Note that Aristotle agrees with Zeno that a moving thing does in fact come into contact with infinite things (i.e. subintervals of motion) in a finite time. For Aristotle it is still counted as an infinity of division because it doesn’t come into contact with anything that is quantitatively infinite. As a result, for Aristotle, this is an infinity of division, which he seems to classify as a potential infinity.

I have no problem with any of this. But my main point is this: By Aristotle’s standards as laid out in his discussion of the dichotomy argument, a convergent series of movers would also be classified as an infinity of division, and therefore a potential infinity, which is therefore possible. In both Aristotle’s and Aquinas’ arguments from motion, they never address this possibility, and so there is a hole in the argument.

It’s absolutely true that at the quantum level that some forms of motion are discontinuous. Quantum tunneling is a famous example in which a particle moves past a barrier that it cannot move through continuously just by vanishing and appearing on the other side.

However, this is only some kinds of motion, not all. On the quantum level ordinary motions can very well be continuous as well. Take for example the motion of a photon. A photon isn’t like a little billiard ball made out of light, but is a packet of an electro-magnetic wave that propagates throughout space continuously at the speed of light.

These kinds of properties aren’t just true for photons, though. All particles have a wave-like aspect to them, and these waves propagate continuously throughout space.

But unlike Aristotle or Aquinas, you are treating convergent series as an instance of an actual infinity. Otherwise your argument does not work, right?

God bless,
Ut

 

[Bakmoon]

But unlike Aristotle or Aquinas, you are treating convergent series as an instance of an actual infinity. Otherwise your argument does not work, right?

God bless,
Ut

No. Although each subinterval of a convergent series of motion is present in actuality rather than in potentiality, and the number of subintervals is infinite, I think that the term actual infinity should only be understood to apply to infinite magnitudes, and since a convergent series doesn’t have an infinite magnitude, it isn’t an actual infinite.

Perhaps it is confusing because in my modified form of the dichotomy argument I talk about it as an actual infinity, but the whole point of my modified dichotomy argument is to give a reductio ad absurdum argument against many people’s insufficient definitions of the potential-actual distinction with regards to infinities.

A lot of people I have talked with over the years often hold very strongly to the idea that all infinite series are automatically actual infinities and refuse to accept my proposal that the distinction between an actual and potential infinity to Aristotle involves a discussion of magnitude, so I use the modified dichotomy argument to force them to see that their definition of an actual infinity is too crude and needs to be revised otherwise motion would be impossible.

 

[Jaaanosik]

It simply does not logically follow that if energy is quantized then motion or space must be simply because energy and motion are related. The actual energy involved in a motion must be quantized but that doesn’t mean that somehow the actual motion itself is discontinuous.

It’s about units of measure in physics. What is the smallest length in what uom and the smallest time in what uom that we can claim that a particle moved? If there is no smallest tick then there is no motion.

An infinitely small point in physics does not exist, it’s a mathematical concept.
A point in physics has to have an energy, dimensions, a motion with respective units of measure otherwise it’s not a point and it does not exist.
A physical point moves when there is a change of the point position in time and the position and time have their units of measure.

Please, define your physical continuous motion.

 

[Linusthe2nd]

Don’t you guys think you are beating a dead horse? Aristotle’s proof for the impossibility of an infinite regress may be good or bad. The point is that few people seem to understand it. Aquinas didn’t object to it, but it is clear that some ( cannot follow the argument, it just doesn’t register. ). On the other hand Thomas, in SCG, Book 1, ch 13 para 14 & 15 gave proofs that can be easily understood. So the proof for the existence of an Unmoved Mover does not rest on Aristotle’s argument. Indeed, common sense tells us that one cannot have a per se ordered series of movers and moved or causes and effects. No one has to be a genius to understand that. One must come to an absolute First which simply IS.

Linus2nd