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Our
natural laws (such as Ohm’s law) are equations expressing ratios of
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measurables
that show consistent experimental connection. A measurable
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divided
by another measureable forms a natural law operator. The
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measurements
are in terms of dimensional numbers that aren’t the numbers of
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pure
mathematics. The natural law
operators are special kinds of
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dimensional
numbers.
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The
numbers mathematicians think of are derived from axioms, and
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are not
based on measurement or dimensional reasoning at all.
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Mathematicians,
as a group, haven’t thought seriously about the arithmetic of
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natural
laws. Perhaps they haven’t felt
there could be a problem with that
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arithmetic.. We have to do arithmetic with
natural laws as we model. But that
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arithmetic
doesn’t have the same provable foundations as arithmetic in pure
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math.
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