[FOM] Re: On the Nature of Mathematical Objects

Timothy Y. Chow tchow at alum.mit.edu
Mon Aug 9 13:02:14 EDT 2004

Dmytro Taranovsky <dmytro at mit.edu> wrote:
> One can try to define what a star is by noting some properties that all
> stars have (such as being heavy) and some properties that no star has,
> but the definition will be in some cases ambiguous (for example, are
> brown dwarfs stars? what about neutron stars?).  By contrast, a
> mathematical definition, such as that of an even integer, has no
> ambiguity.

So for example, if

       / the empty set,           if the continuum hypothesis holds, and
  x = <
       \ the class of all sets,   otherwise,

then it is unambiguous whether x is a set?

> The reach of particular physical objects is ambiguous (for
> example, is solar corona a part of the sun?), but no such ambiguity
> exists, say, for the set of prime numbers:  The set reaches those only
> those natural numbers that have exactly two different divisors.

And no ambiguity exists about whether x (as defined above) "reaches," say,
the set of all integers?

Whether you answer yes or no to these questions, it seems that the answer
is far from uncontroversial.


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