Georg Cantor (1845-1918)
Georg Cantor "defined" the natural number system from the empty set {} or Ø. This is associated with the number 0.
You can then associate {Ø} with the number 1, { Ø ,{Ø}} with 2 and so on.
Penrose [1] points argues that this shows how numbers can be conjured out of nothing, and that "we get an infinite sequence of abstract (Platonic) mathematical entities" independent of physical nature.
What he misses, in my opinion, is that the natural numbers are not generated from the empty set as a flower might be generated from a seed. The natural numbers are linked to Cantors sequence of sets like a row of diplomats shaking hands. The diplomats on both sides have to exist before the hand shaking can begin, i.e. the natural numbers have to exist before you can associate them with sets. And the only way I can conceive of obtaining the natural numbers is by perceiving them from
reasonably well-defined discrete objects that persist in time.
Georg Cantor timeline:
- Georg Cantor
- Roger Penrose
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