[FOM] Real Numbers
Sandy Hodges
SandyHodges at attbi.com
Wed May 14 17:01:34 EDT 2003
Lucas Wiman writes:
When mathematicians (myself included) say equal or the same, we often
mean isomorphic. Philosophers frequently miss this point.
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By any definition of numbers I know of, the natural number 3 is not
identical to the rational number 3, and neither is identical to the real
number 3. A sentence using "3" that does not specify which 3 is
ambiguous. Such texts can rarely be fixed by translating "same" as
"isomorphic to". There is of course not the least reason to keep
track of what sort of number any particular use of "3" represents.
Nevertheless, a text with an ambiguous "3" is not strictly rigorous.
There is and can be no "mathematical way of speaking" that makes the
difference between the various 3s cease to exist, however harmless it
may be to ignore the distinction almost all of the time.
Similarly, the real 3 that is a Dedekind Cut is not identical to the
real 3 that is an equivalence set of Cauchy sequences. Two number
systems may indeed be isomorphic, a given number has to be one or the
other. (And even if there is a Platonic real number systems, to which
the various defined real systems are merely isomorphic, then a Platonic
3 is a Platonic 3, and it is not a Dedekind 3.)
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Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.
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