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It was pure conjecture by the Greeks.
No purely conjecture. If something ground down to finer and finer particles, it's fair to assume that matter comes down to some fundamental particle as a building 'block' - their proposed "atomos."
Why?
Why would you assume that matter isn't continuous?
I can cut a lump of wax in half, and end up with two lumps of wax. Why would you assume that I couldn't keep doing that indefinitely (assuming the ability to manipulate arbitrarily small things)?
I don't think that's a 'fair assumption' at all.
It feels true to us, because we have grown up with the knowledge that it is true. But in the absence of that upbringing, it seems like it would be a highly counterintuitive idea.
"I can cut this in half thirty six times, but not thirty seven" sounds like the ranting of an idiot, in the absence of any experimental support for the claim. Particularly given that there's no obvious limit to how much stuff you can add - if I have two similar lumps of wax, I can join them to make one larger lump, and that's repeatable indefinitely, given a sufficient supply of wax. Why on Earth would it be intuitively obvious that the reverse process cannot continue indefinitely?
Yet they didn't, the Greeks stopped at 'atomos,' which they assumed were solid little balls of fundamental material.
Indivisible, immutable, and of universal inertia. Matter had atomos, but so did things like perception, the soul, and the color blue.
In essence, a material basis for a material world even though they got many of the details wrong, including the nature of atoms?
What we call 'atoms' today aren't much like what they meant by 'atomos'.
The idea was that they would be the smallest indivisible parts of anything, but that description is analogous to many different levels of subdivision that are recognised today.
If you cut a piece of wood into smaller and smaller parts, when does it stop being a piece of wood?
Based on the modern understanding of what constitutes 'wood', the ability to divide further without the products being 'wood' probably comes at the level where at least one of every different molecule found in wood is still present.
Another candidate for 'atomos' is the single cellulose molecule, though that's arguably not 'wood' anymore, but a mere component of wood. If you collect together only pure cellulose, you cannot make it into wood - you need traces of other molecules (mainly, but not exclusively, water).
If we allow subdivision to the molecule level, and ignore that we are no longer strictly dealing with 'wood', then we can go deeper - Cellulose can be broken down into (modern) atoms, of carbon, hydrogen and oxygen.
But why stop there? If a molecule isn't 'atomos', why is an atom? We can keep cutting, to get electrons, protons and neutrons.
And those can be further divided into leptons and hadrons.
Which can be divided into quarks.
We don't know whether further divisions are possible yet; It all gets a lot complicated at this scale. But modern physics describes everything as fields, or strings, or other continuous objects that at least hypothetically can be divided indefinitely.
The question of whether reality is continuous at arbitrarily small scales isn't resolved; And according to Planck, it may be impossible to ever observe scales below a certain limit, which may or may not imply that objects below that size don't exist at all.
Despite our modern adoption of 'atom' to mean the smallest possible amount of a given chemical element, we have yet to determine whether an uncuttable 'atomos' is a reality; We can, however, say with absolute confidence that 'atoms' aren't it - just ask the people of Hiroshima.
They didn't "get many of the details wrong"; We have arbitrarily stolen their word and used it to mean something rather different. If anything, it's the modern word 'atom' that's wrong.
The Greek philosophers discussing 'atomos' weren't wrong, they were talking about something completely different. In much the same way that the old understanding of 'elements' isn't a wrong description of chemical elements as we understand them today; It's a very good description of physical states of matter, with some (incorrect) speculation that these physical states are the fundamental determinants of the chemical properties of macroscopic materials.
Earth, air, fire and water are simply an archaic way to say solid, gas, plasma and liquid; I think we can reasonably forgive our forebears for not knowing about Bose-Einstein condensates.