> If there were no smallest-possible particle, and everything were infinitely divisible, then a grain of sand would contain infinite particles. The entire universe would also contain infinite particles. But the universe and a grain of sand are clearly not the same size — they’re not both infinite. Therefore, there must be some smallest, indivisible particle. An “atom.”
This is a very bad argument. It also "proves" that space and time are made of "atoms". Which may or may not be true, we don't know. But this argument doesn't help us to know.
It's a bad argument because two infinities don't have to be the same size. Consider e.g. the whole numbers vs the odd numbers. Both infinite, yet one is contained within the other.
Don't get me wrong! Democritus, Epicurus, Lucretius seem to me to have had the right of it on very many things. But I can see why this argument wouldn't have convinced anyone who didn't already believe in atoms. It's like the various mediaeval proofs of the existence of God. Convincing only if you already believe the conclusion.
>> The argument above, despite being clever, is also wrong. ChatGPT-4 explains the logical error involved: “Infinite divisibility does not imply infinite quantity: The argument assumes that if something is infinitely divisible, it must contain an infinite number of particles. However, infinite divisibility refers to the theoretical ability to divide something indefinitely, not the actual presence of an infinite number of particles. For example, a line segment can be divided into infinitely smaller segments, but this does not mean the line segment contains an infinite number of [size-inflexible] segments.”]
I disagree with ChatGPT-4; any line segment can indeed be considered to contain arbitrarily many segments, [size-inflexible] is begging the question.
Really, the error is the same made by the commenter above: who says that a grain of sand is smaller than the universe? Surely, a bijective relation is possible. Similarly, while two infinities are not necessarily the same size, the sets of whole numbers and odd numbers are!
You might wish to investigate the opponents to both the Epicureans and the Stoics: the Pyrrhonists. The Pyrrhonists agree with you that "you can’t find any “good” or “bad” written into the universe." Their complaint with the Epicureans and particularly the Stoics is that they came to rash conclusions in their doctrines and that there was no way to tell whether those conclusions were correct, especially given how they were contradicted by the doctrines of other schools.
Thank you for a brief, condensed exposure to Philosophy. A lovely and unexpected surprise from a statistician. more correctly a statistical analyzer of data. Amazing that someone living in those times could think and write like this, when the main goal every morning upon arising for most people was "what can I find to eat today or tomorrow?"
The argument for atoms could be used to "prove" that the amount of real numbers in an interval is non-infinite, because the "size" of 1..2 is smaller than 1..10. So the argument isn't sound.
Still interesting that he had a rational approach to these questions, thanks for sharing.
Infinite numbers are certainly pretty hard to get one's head around. It's intuitive that the "size" of the real numbers within 1..2 should be smaller than 1..10. After all, there are numbers within 1..10 which we can identify that definitely aren't in 1..2 (and not vice versa.) Kind of seems like it's bigger ... but I realize the limitation is in my inability to grasp infinity here.
> If there were no smallest-possible particle, and everything were infinitely divisible, then a grain of sand would contain infinite particles. The entire universe would also contain infinite particles. But the universe and a grain of sand are clearly not the same size — they’re not both infinite. Therefore, there must be some smallest, indivisible particle. An “atom.”
This is a very bad argument. It also "proves" that space and time are made of "atoms". Which may or may not be true, we don't know. But this argument doesn't help us to know.
I don't know if I'd call it "very bad". It's certainly clever, and I still haven't put my finger on exactly why it's wrong, though I'm sure it is.
It's a bad argument because two infinities don't have to be the same size. Consider e.g. the whole numbers vs the odd numbers. Both infinite, yet one is contained within the other.
Don't get me wrong! Democritus, Epicurus, Lucretius seem to me to have had the right of it on very many things. But I can see why this argument wouldn't have convinced anyone who didn't already believe in atoms. It's like the various mediaeval proofs of the existence of God. Convincing only if you already believe the conclusion.
Thanks.
I added a note to the article:
>> The argument above, despite being clever, is also wrong. ChatGPT-4 explains the logical error involved: “Infinite divisibility does not imply infinite quantity: The argument assumes that if something is infinitely divisible, it must contain an infinite number of particles. However, infinite divisibility refers to the theoretical ability to divide something indefinitely, not the actual presence of an infinite number of particles. For example, a line segment can be divided into infinitely smaller segments, but this does not mean the line segment contains an infinite number of [size-inflexible] segments.”]
I disagree with ChatGPT-4; any line segment can indeed be considered to contain arbitrarily many segments, [size-inflexible] is begging the question.
Really, the error is the same made by the commenter above: who says that a grain of sand is smaller than the universe? Surely, a bijective relation is possible. Similarly, while two infinities are not necessarily the same size, the sets of whole numbers and odd numbers are!
Very fun and thought provoking write up, thanks for sharing.
Thank you!
You might wish to investigate the opponents to both the Epicureans and the Stoics: the Pyrrhonists. The Pyrrhonists agree with you that "you can’t find any “good” or “bad” written into the universe." Their complaint with the Epicureans and particularly the Stoics is that they came to rash conclusions in their doctrines and that there was no way to tell whether those conclusions were correct, especially given how they were contradicted by the doctrines of other schools.
Interesting! Thanks for the tip
I'll put in a plug for my own book on the topic: https://www.amazon.com/Pyrrhos-Way-Ancient-Version-Buddhism/dp/1896559565/ref=sr_1_1
Thank you for a brief, condensed exposure to Philosophy. A lovely and unexpected surprise from a statistician. more correctly a statistical analyzer of data. Amazing that someone living in those times could think and write like this, when the main goal every morning upon arising for most people was "what can I find to eat today or tomorrow?"
The argument for atoms could be used to "prove" that the amount of real numbers in an interval is non-infinite, because the "size" of 1..2 is smaller than 1..10. So the argument isn't sound.
Still interesting that he had a rational approach to these questions, thanks for sharing.
Thanks.
Infinite numbers are certainly pretty hard to get one's head around. It's intuitive that the "size" of the real numbers within 1..2 should be smaller than 1..10. After all, there are numbers within 1..10 which we can identify that definitely aren't in 1..2 (and not vice versa.) Kind of seems like it's bigger ... but I realize the limitation is in my inability to grasp infinity here.