matter and objects seem like the real thing because we evolved to interact with the world and these are graspable ideas. But they do fall apart as you've argued in the beginning of your article. A can could be defined by its atoms but its not the atoms. It could be defined by its form but that form can exist a million times. can we overcome this problem by defining it in space and time? but ill agree that Forms and matter have a hard time defining things as the matter and form can change and still be considered the same thing. Replace every plank on the ship...
Tell me if this idea of possibilities is related to Whiteheads ideas of processes. It seems better to define an object as a process if its growing or eroding or moving. These exist across time rather than like a snapshot of the current state. We try to tie down the self , processes seem to give us a better definition than a current state. The continuity of being holds a thing together despite all parts changing. I like this definition but it also leads to the process of ME is just part of every other process in the world. I am a part of the process of Chicken that I eat. I am a part of AIr and WIND that i breath and move through my blood. So in the end there is only one process and that doesnt help us as far as i can see.
Perhaps your idea of possabilaties and whiteheads ideas of processes are better in some senses but they are hard to grasp when i look out my window and try to count things. What counts as a thing?
Lots of questions and not many good answers but thats why philosophy is engaging
Bringing in space and time seems like a good move, although I suppose then we're almost abandoning the idea of the thing persisting across time... But it's an important aspect of a thing in any case.
My ideas here are pretty similar to Whitehead's, although he doesn't speak about processes in quite the way you are, at least from what I've read of him so far. The Platonic Forms are essentially the same as Whitehead's "eternal objects", except Whitehead recognised the eternal objects as being pure possibilities. His understanding of things' persisting/changing was in terms of "societies" of "actual entities" (the fundamental things of reality, existing relationally with each other) spread across time, all partaking of a common eternal object, which is realized in their mutual relations to each other (eg the actual entities that exist within in a ball are all related to each other in a way that realizes the eternal object "ball").
To consider the One the ultimate abstraction is perfectly dualistic. Nondualism is the polar opposite solution : the One (Tao) is unabstracted knowledge (consciousness), known before conceiving yet a mystery once conveived/abstracted
So this is actually a good argument for the One, who is the ground of the actuality of all things.
Yes, matter per se is but possibility, and likewise so are the forms. Then whither the actuality of all the things we see? They are not actual in virtue of matter, nor are they actual in virtue of their form. Therefore their actuality comes from that which underlies both matter and form, and that is the One, whom the middle Platonists identified as the First God, the Father of all things.
The One doesn't give any information to those approaching the divine essence from the perspective of matter and form. Obviously so, for he is neither matter nor form. But it doesn't follow from this that the One is purely possibility — rather, he is, as Philo of Alexandria stated, Being itself. And Being is not indeterminate in itself, for it stands in opposition to nonbeing. What makes matter and form what they are is their delimitation. Individuals themselves and the forms underlying them are delimited by being _these_ ways of being actual but not _those_ ways. But the One is infinite actuality itself, and infinity is not indeterminate in itself, it appears so only if one assumes that what exists but exist in a delimited way.
>But it doesn't follow from this that the One is purely possibility — rather, he is, as Philo of Alexandria stated, Being itself.
What if Being itself is pure possibility? The name, "Being itself" already reveals that we are abstracting away all the instances of beings. Does it not make sense to see "Being itself" as a kind of blank space offering the possibility for beings?
>And Being is not indeterminate in itself, for it stands in opposition to nonbeing.
Does it stand in opposition to non-being though? Lots of platonists, from Plato himself through to Pseudo-Dionysus, have understood the One as beyond both being and non-being.
When you say that the One is "infinite actuality itself", what do you mean by "actuality"?
When we say "Being itself" we are abstracting away all the delimitations of finite beings. That doesn't by itself entail that Being is _only_ possibility. Being is both possibility and actuality, because whatever is actual is also possible.
If Being is beyond both being and non-being then it does stand in opposition to non-being. There are multiple ways in which "beyond being and non-being" is parsed in the literature, if by this one means Being neither exists not doesn't exist that is absurd, but if by this one understands that Being is beyond _ousia_, substance then that's perfectly licit. Being is not a substance as we know substances.
I don't think actuality can be defined, but since you speak of possibility, you already grasp what actuality is: every possibility is the possibility of some actuality.
I hadn't thought about the idea of matter (atoms) being an abstraction as opposed to a concrete basis of reality before. It makes me think of Euclidean Geometry. In Euclidean Geometry, one usually thinks of points as being fundamental, and lines being a set of points arranged in "linear fashion". However, we don't have to think of it this way: instead, we could think of the lines as being fundamental, and then say that a point is a set of lines arranged as if "rotated in a circle". I suspect you could build an isomorphic Geometry where lines are primary (and for projective geometries points and lines are literally interchangeable). If you were a creature living in Euclidean Geometry, you might not even be able to tell whether lines or points are fundamental because of this isomorphism.
Similarly, in the real world there may be multiple, isomorphic ways of building reality. For example, maybe there is a version where waves within quantum fields are fundamental, and another version where particles are fundamental and just exhibit wave-like properties. It may be impossible for us to tell which is the real story. However, there must *be* a real story. If the universe was agnostic between the options, I don't see in what sense anything would exist. This is similar how there may be many equivalent ways to get to my vacation destination, but I actually have to select one if I really want to get there.
Yes, excellent point about points! This is exactly how Aristotle thought about both matter and the relationship between points and lines. He saw that points are not fundamental, but only exist as potentialities within a line - places where we might make a cut. This was how he resolved Zeno's racetrack paradox - a distance has infinite *potential* for division, but that doesn't mean it's *actually* infinitely divided, which is, for Aristotle, impossible (I wrote about this in a post on my old blog if you're interested - https://asalittlechild.wordpress.com/2024/06/02/time-is-not-made-up-of-instants-zenos-dichotomy-paradox/). Points are the matter of a line or region.
>Similarly, in the real world there may be multiple, isomorphic ways of building reality.
Personally, I think there are multiple isomorphic ways of *looking at* reality, but that reality itself does not have any preference between them. So I suppose I am denying that there is a "real story". Or we could say that the real story is just reality, without all our conceptualisations of it. Reality just is. All our concepts are just ways of cutting the cake that is the whole of reality.
Thanks for the link to the article about chopping up time! I do disagree about a mathematical point- not that your math is wrong, but I disagree philosophically. Sorry this got really long, feel free to ignore the next paragraph!
You say "An infinite series is not the sum of infinitely many terms." I think it is the sum of infinitely many terms. It's certainly not a finite sum. You could say it's actually not a sum at all (more on that below), but traditionally I think it is seen as an actual infinite sum. Now your description of limits is correct. So for example to find 1/3 + 1/9 + 1/27 + ..., we'd take an arbitrary epsilon > 0, and find an n such that the finite sum 1/3 + 1/9 + ... + 1/3^n is within epsilon of the final answer of 1/2. Here epsilon and n are finite values, as you say. But they aren't specific finite values like 1/1000 or 50,203. The argument applies in generality. That is, the argument applies to a true infinite set of values of epsilon and n. As a result, this shows the original infinite sum, which really does have an infinite number of terms, cannot differ from 1/2. Now, you could say that an "infinite sum" is not a sum at all and is really this process of coming up with a finite, symbolic argument satisfying the definition of a limit. But I think the original intention was that the infinite sum is what it is, a sum with infinite terms, and the limit is just a tool we finite humans need to resort to in order to prove rigorously what the value is, since we can't actually add up all the terms.
Having said that, I agree with your larger point about Zeno's paradox. Anyone who goes deep into learning about the real numbers I think will have reasons to be skeptical that they really exist at all (Banach-Tarski, continuum hypothesis, etc.). I think resolving Zeno's paradox without resorting to real numbers is a great idea, and you've done that and explained it very well.
Thanks for reading it, and engaging with the maths/philosophy! I love this stuff, and especially don't get to discuss the mathematical points enough.
>"The argument applies in generality. That is, the argument applies to a true infinite set of values of epsilon and n. As a result, this shows the original infinite sum, which really does have an infinite number of terms, cannot differ from 1/2."
Good point that it applies to an infinite set of values of epsilon and n. But that is still an infinite set of *finite* sums! We have not justified the leap from this infinite set of finite sums to a single infinite sum. I do not believe we can, but perhaps one day it will be proven.
But also, seeing it as a single infinite sum causes very serious issues (and probably even a contradiction) if we look at conditionally convergent series, since we can show that these can converge on any value at all if the terms of the series are rearranged properly. If we were actually adding together all the infinite terms, this should not be possible, right? Otherwise, we could show that 1=0 by rearranging a conditionally convergent series appropriately.
>I think resolving Zeno's paradox without resorting to real numbers is a great idea, and you've done that and explained it very well.
Yes! I was thinking of quantum states as I wrote it, but didn't want to go off on a big tangent. But it really does seem like quantum indeterminacy is the present exploring possibility space (the future?) before it makes its choices and becomes the past.
Interesting post. I might not have caught every point, but here are my thoughts.
My criteria for what is real is usually long the lines of the Eleatic dictum, that what is causal is real. Although more fundamentally, that would be what is structural and related to other real things. Maybe another way of saying it is, what is real is what we can only ignore with a cost, or a risk of a cost.
I'm not sure about a "case of misplaced concreteness." I do know that with modern physics, as we drill down into concreteness, we find it's more about arrangements that resist penetration through forces, in other words, processes.
I'm not enthusiastic for Platonic Forms, but Aristotelian forms, as in universal functionalism, is a different matter. Note the above about concreteness being about resisting penetration, in other words, a functional role.
I think possibility exists in our ignorance, but at least some of that ignorance may be unresolvable, at least ahead of time. So we have no choice but to deal with possibilities. Ignoring them has a cost.
>My criteria for what is real is usually long the lines of the Eleatic dictum, that what is causal is real. Although more fundamentally, that would be what is structural and related to other real things. Maybe another way of saying it is, what is real is what we can only ignore with a cost, or a risk of a cost.
I am very much on board with the Eleatic principle. The difficulty, I think, is that it's not necessarily clear what we mean by a "cause". Does it include all of Aristotle's four causes? Can an absence, such as a hole in a bucket, have causal powers? And what if the cause can be explained just as well in terms of a different kind of causality, as with reductionism? It is tricky... Perhaps we could count the cost of reductionism in the computational costs of actually using the "reduced" explanation?
>I'm not sure about a "case of misplaced concreteness." I do know that with modern physics, as we drill down into concreteness, we find it's more about arrangements that resist penetration through forces, in other words, processes.
Sorry, I think I expressed myself poorly there. I wasn't referring to the property of solidity, but to Whitehead's notion of the 'fallacy of misplaced concreteness', where we mistake what is really an abstraction (in this case I'm claiming matter is an abstraction) for what's really real in the concrete world we live in. I'll go and reword that bit, because it was not clear what I meant at all, sorry.
>I'm not enthusiastic for Platonic Forms, but Aristotelian forms, as in universal functionalism, is a different matter.
I had been much more favourable to Aristotelian forms too, but I think they fall short because they are basically the second answer I mentioned in the post, that concepts are only real as substructures of concrete reality. I think the third answer, that they exist as part of a possibility space, is much more Platonist. Although Aristotle did have some interesting ideas about mathematics as being concerned with potentialities, so perhaps there's some potential convergence there.
Im interested in the Platoinic realm and forms, Ill have to reread your piece to get a better handle on it
Nice, let me know if you've got any thoughts or questions on it :)
matter and objects seem like the real thing because we evolved to interact with the world and these are graspable ideas. But they do fall apart as you've argued in the beginning of your article. A can could be defined by its atoms but its not the atoms. It could be defined by its form but that form can exist a million times. can we overcome this problem by defining it in space and time? but ill agree that Forms and matter have a hard time defining things as the matter and form can change and still be considered the same thing. Replace every plank on the ship...
Tell me if this idea of possibilities is related to Whiteheads ideas of processes. It seems better to define an object as a process if its growing or eroding or moving. These exist across time rather than like a snapshot of the current state. We try to tie down the self , processes seem to give us a better definition than a current state. The continuity of being holds a thing together despite all parts changing. I like this definition but it also leads to the process of ME is just part of every other process in the world. I am a part of the process of Chicken that I eat. I am a part of AIr and WIND that i breath and move through my blood. So in the end there is only one process and that doesnt help us as far as i can see.
Perhaps your idea of possabilaties and whiteheads ideas of processes are better in some senses but they are hard to grasp when i look out my window and try to count things. What counts as a thing?
Lots of questions and not many good answers but thats why philosophy is engaging
Bringing in space and time seems like a good move, although I suppose then we're almost abandoning the idea of the thing persisting across time... But it's an important aspect of a thing in any case.
My ideas here are pretty similar to Whitehead's, although he doesn't speak about processes in quite the way you are, at least from what I've read of him so far. The Platonic Forms are essentially the same as Whitehead's "eternal objects", except Whitehead recognised the eternal objects as being pure possibilities. His understanding of things' persisting/changing was in terms of "societies" of "actual entities" (the fundamental things of reality, existing relationally with each other) spread across time, all partaking of a common eternal object, which is realized in their mutual relations to each other (eg the actual entities that exist within in a ball are all related to each other in a way that realizes the eternal object "ball").
Indeed, there's always more to ponder!
And possibilities are just that, possibilities, which contradict reality
To consider the One the ultimate abstraction is perfectly dualistic. Nondualism is the polar opposite solution : the One (Tao) is unabstracted knowledge (consciousness), known before conceiving yet a mystery once conveived/abstracted
A perhaps relevant quote: "when the doors of perception are cleansed, things are seen as they are; infinite" -- William Blake
So this is actually a good argument for the One, who is the ground of the actuality of all things.
Yes, matter per se is but possibility, and likewise so are the forms. Then whither the actuality of all the things we see? They are not actual in virtue of matter, nor are they actual in virtue of their form. Therefore their actuality comes from that which underlies both matter and form, and that is the One, whom the middle Platonists identified as the First God, the Father of all things.
The One doesn't give any information to those approaching the divine essence from the perspective of matter and form. Obviously so, for he is neither matter nor form. But it doesn't follow from this that the One is purely possibility — rather, he is, as Philo of Alexandria stated, Being itself. And Being is not indeterminate in itself, for it stands in opposition to nonbeing. What makes matter and form what they are is their delimitation. Individuals themselves and the forms underlying them are delimited by being _these_ ways of being actual but not _those_ ways. But the One is infinite actuality itself, and infinity is not indeterminate in itself, it appears so only if one assumes that what exists but exist in a delimited way.
>But it doesn't follow from this that the One is purely possibility — rather, he is, as Philo of Alexandria stated, Being itself.
What if Being itself is pure possibility? The name, "Being itself" already reveals that we are abstracting away all the instances of beings. Does it not make sense to see "Being itself" as a kind of blank space offering the possibility for beings?
>And Being is not indeterminate in itself, for it stands in opposition to nonbeing.
Does it stand in opposition to non-being though? Lots of platonists, from Plato himself through to Pseudo-Dionysus, have understood the One as beyond both being and non-being.
When you say that the One is "infinite actuality itself", what do you mean by "actuality"?
When we say "Being itself" we are abstracting away all the delimitations of finite beings. That doesn't by itself entail that Being is _only_ possibility. Being is both possibility and actuality, because whatever is actual is also possible.
If Being is beyond both being and non-being then it does stand in opposition to non-being. There are multiple ways in which "beyond being and non-being" is parsed in the literature, if by this one means Being neither exists not doesn't exist that is absurd, but if by this one understands that Being is beyond _ousia_, substance then that's perfectly licit. Being is not a substance as we know substances.
I don't think actuality can be defined, but since you speak of possibility, you already grasp what actuality is: every possibility is the possibility of some actuality.
I hadn't thought about the idea of matter (atoms) being an abstraction as opposed to a concrete basis of reality before. It makes me think of Euclidean Geometry. In Euclidean Geometry, one usually thinks of points as being fundamental, and lines being a set of points arranged in "linear fashion". However, we don't have to think of it this way: instead, we could think of the lines as being fundamental, and then say that a point is a set of lines arranged as if "rotated in a circle". I suspect you could build an isomorphic Geometry where lines are primary (and for projective geometries points and lines are literally interchangeable). If you were a creature living in Euclidean Geometry, you might not even be able to tell whether lines or points are fundamental because of this isomorphism.
Similarly, in the real world there may be multiple, isomorphic ways of building reality. For example, maybe there is a version where waves within quantum fields are fundamental, and another version where particles are fundamental and just exhibit wave-like properties. It may be impossible for us to tell which is the real story. However, there must *be* a real story. If the universe was agnostic between the options, I don't see in what sense anything would exist. This is similar how there may be many equivalent ways to get to my vacation destination, but I actually have to select one if I really want to get there.
Yes, excellent point about points! This is exactly how Aristotle thought about both matter and the relationship between points and lines. He saw that points are not fundamental, but only exist as potentialities within a line - places where we might make a cut. This was how he resolved Zeno's racetrack paradox - a distance has infinite *potential* for division, but that doesn't mean it's *actually* infinitely divided, which is, for Aristotle, impossible (I wrote about this in a post on my old blog if you're interested - https://asalittlechild.wordpress.com/2024/06/02/time-is-not-made-up-of-instants-zenos-dichotomy-paradox/). Points are the matter of a line or region.
>Similarly, in the real world there may be multiple, isomorphic ways of building reality.
Personally, I think there are multiple isomorphic ways of *looking at* reality, but that reality itself does not have any preference between them. So I suppose I am denying that there is a "real story". Or we could say that the real story is just reality, without all our conceptualisations of it. Reality just is. All our concepts are just ways of cutting the cake that is the whole of reality.
Thanks for the link to the article about chopping up time! I do disagree about a mathematical point- not that your math is wrong, but I disagree philosophically. Sorry this got really long, feel free to ignore the next paragraph!
You say "An infinite series is not the sum of infinitely many terms." I think it is the sum of infinitely many terms. It's certainly not a finite sum. You could say it's actually not a sum at all (more on that below), but traditionally I think it is seen as an actual infinite sum. Now your description of limits is correct. So for example to find 1/3 + 1/9 + 1/27 + ..., we'd take an arbitrary epsilon > 0, and find an n such that the finite sum 1/3 + 1/9 + ... + 1/3^n is within epsilon of the final answer of 1/2. Here epsilon and n are finite values, as you say. But they aren't specific finite values like 1/1000 or 50,203. The argument applies in generality. That is, the argument applies to a true infinite set of values of epsilon and n. As a result, this shows the original infinite sum, which really does have an infinite number of terms, cannot differ from 1/2. Now, you could say that an "infinite sum" is not a sum at all and is really this process of coming up with a finite, symbolic argument satisfying the definition of a limit. But I think the original intention was that the infinite sum is what it is, a sum with infinite terms, and the limit is just a tool we finite humans need to resort to in order to prove rigorously what the value is, since we can't actually add up all the terms.
Having said that, I agree with your larger point about Zeno's paradox. Anyone who goes deep into learning about the real numbers I think will have reasons to be skeptical that they really exist at all (Banach-Tarski, continuum hypothesis, etc.). I think resolving Zeno's paradox without resorting to real numbers is a great idea, and you've done that and explained it very well.
Thanks for reading it, and engaging with the maths/philosophy! I love this stuff, and especially don't get to discuss the mathematical points enough.
>"The argument applies in generality. That is, the argument applies to a true infinite set of values of epsilon and n. As a result, this shows the original infinite sum, which really does have an infinite number of terms, cannot differ from 1/2."
Good point that it applies to an infinite set of values of epsilon and n. But that is still an infinite set of *finite* sums! We have not justified the leap from this infinite set of finite sums to a single infinite sum. I do not believe we can, but perhaps one day it will be proven.
But also, seeing it as a single infinite sum causes very serious issues (and probably even a contradiction) if we look at conditionally convergent series, since we can show that these can converge on any value at all if the terms of the series are rearranged properly. If we were actually adding together all the infinite terms, this should not be possible, right? Otherwise, we could show that 1=0 by rearranging a conditionally convergent series appropriately.
>I think resolving Zeno's paradox without resorting to real numbers is a great idea, and you've done that and explained it very well.
Thank you :)
I think Plato's forms are closest to being real as mathematical abstractions. The possibilities bring to mind quantum states.
Yes! I was thinking of quantum states as I wrote it, but didn't want to go off on a big tangent. But it really does seem like quantum indeterminacy is the present exploring possibility space (the future?) before it makes its choices and becomes the past.
Interesting post. I might not have caught every point, but here are my thoughts.
My criteria for what is real is usually long the lines of the Eleatic dictum, that what is causal is real. Although more fundamentally, that would be what is structural and related to other real things. Maybe another way of saying it is, what is real is what we can only ignore with a cost, or a risk of a cost.
I'm not sure about a "case of misplaced concreteness." I do know that with modern physics, as we drill down into concreteness, we find it's more about arrangements that resist penetration through forces, in other words, processes.
I'm not enthusiastic for Platonic Forms, but Aristotelian forms, as in universal functionalism, is a different matter. Note the above about concreteness being about resisting penetration, in other words, a functional role.
I think possibility exists in our ignorance, but at least some of that ignorance may be unresolvable, at least ahead of time. So we have no choice but to deal with possibilities. Ignoring them has a cost.
No idea on Western philosophy and emptiness.
>My criteria for what is real is usually long the lines of the Eleatic dictum, that what is causal is real. Although more fundamentally, that would be what is structural and related to other real things. Maybe another way of saying it is, what is real is what we can only ignore with a cost, or a risk of a cost.
I am very much on board with the Eleatic principle. The difficulty, I think, is that it's not necessarily clear what we mean by a "cause". Does it include all of Aristotle's four causes? Can an absence, such as a hole in a bucket, have causal powers? And what if the cause can be explained just as well in terms of a different kind of causality, as with reductionism? It is tricky... Perhaps we could count the cost of reductionism in the computational costs of actually using the "reduced" explanation?
>I'm not sure about a "case of misplaced concreteness." I do know that with modern physics, as we drill down into concreteness, we find it's more about arrangements that resist penetration through forces, in other words, processes.
Sorry, I think I expressed myself poorly there. I wasn't referring to the property of solidity, but to Whitehead's notion of the 'fallacy of misplaced concreteness', where we mistake what is really an abstraction (in this case I'm claiming matter is an abstraction) for what's really real in the concrete world we live in. I'll go and reword that bit, because it was not clear what I meant at all, sorry.
>I'm not enthusiastic for Platonic Forms, but Aristotelian forms, as in universal functionalism, is a different matter.
I had been much more favourable to Aristotelian forms too, but I think they fall short because they are basically the second answer I mentioned in the post, that concepts are only real as substructures of concrete reality. I think the third answer, that they exist as part of a possibility space, is much more Platonist. Although Aristotle did have some interesting ideas about mathematics as being concerned with potentialities, so perhaps there's some potential convergence there.