Why cannot matter be infinitely divisible, according to Plato?
Thanks in advance!
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Eventually, dividing something into smaller and smaller parts causes the thing to lose its emergent properties (i.e., its inherent characteristics that we can observe in it and use to describe it). For example, dividing a cell into subcellular components will cause the cell to lose its emergent properties (e.g., self-replication, RNA translation, homeostasis, metabolism, etc.). Plato specifically refered to further dividing atoms into smaller and smaller parts: an element is defined by the number of protons in its nucleus; if we further divide an atom of Carbon (6 protons), it will lose at least a portion of these protons and become some other type of atom (e.g., maybe now we have two atoms with 3 protons each = Lithium).
Advances in our understanding of science can further assist us in dissecting Plato's point. For instance, Kenneth Wilson, a Nobel Laureate Mathematician/Physicist, defines the correlation length as the smallest amount of something needed for its properties that are apparent in larger amounts to subsist. This conception is analogous to Plato's, although more realistic implications are outlined.
Here is a link to the particular Wilson paper, titled "The renormalization group and the epsilon exapansion": https://pdfs.semanticscholar.org/e372/a0dc3053d0630788bc778dbffd6b4ea5d34b.pdf
Plato actually believed that the most fundamental components of objects were not particulate in nature. To Plato, such a mechanistic model of the world would fail to produce the order and beauty that we see in the natural world. Rather, he believed that at a base level, the elements had faces which were composed of triangles. So, he did believe in a physical theory which incorporated the idea of indivisible components, but these components were mathematical in nature. This is consistent with the views presented in Timaeus wherein he suggests that the cosmos was created according to a model.