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Atomism and Mathematics
It’s possible that Democritus thought not just of matter, but also of
space
in an atomistic way.
That is, the size of an atom would be an
atomic space
. In such a system, the
ultimate unit of
measurement
would be the size of an atom. Within that framework, the very notion of
half
of an
atomic space would be unintelligible. So, Democritus would be able to say, coherently, that an
atom has size even though it is theoretically indivisible.
a.
The
“
Weyl Tile
”
Argument
:
But if Democritus “atomized” space in this way (as he very likely did), he runs into
another problem. For Euclidean geometry (in particular, the Pythagorean theorem)
requires that
space
be continuously divisible. Hence, if atomism denies the continuity of
space, it will fail to get mathematics right.
Why is this? There is a famous argument by the mathematician Hermann Weyl (the
“Weyl tile” argument) that clearly shows what is problematical about atomistic geometry:
Consider any geometrical figure (e.g., squares, triangles, etc.) with straight lines as sides.
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This note was uploaded on 11/14/2011 for the course PHI PHI2010 taught by Professor Jorgerigol during the Fall '09 term at Broward College.
 Fall '09
 JorgeRigol

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