Re: naive question from a non-mathematician

Gene Ward Smith wrote:
Stephen Montgomery-Smith wrote:


Like I said in a different post, it depends upon your framework. Most
modern mathematicians define everything in terms of sets and set theory.

I think that the main point in my post was "it depends upon your framework." But since you seem to like to argue for the sake of arguing, let me keep it going.

Most modern mathematicians define things up to isomorphism.

I don't think that this is as universally understood as you make it out to be. Evidence of this is that if someone says "there are two groups of order 4" they usually add (and especially in an environment where rigor is expected) "up to isomorphism."


Thus the natural numbers are, more or less, defined as the set of
finite ordinals, the integers are pairs of natural numbers quotiented
out by the equivalence relation (a,b)~(c,d) iff a+d=b+c, the rationals
are pairs of integers (the second being non-zero) quotiented out by
another approprate equivalence relation, the reals are constructed from
the rationals usually either by Dedekind sections, or by some quotient
of the cauchy sequences, and the complex numbers are pairs of real numbers.


If you like, you can do things this way. Nobody forces you to, and you
could do it other ways. You could, for instance, define the reals
axiomatically, and recover the rationals and integers from that.


There are just so many ways of defining these objects in set theory, and
so to say R is a subset of C just doesn't cut it if you are going to be
nit-picking.


Let's say I define the complex numbers C as an algebraically closed
field of characteristic zero and cardinality the continuum with a
distinguished automorphism conj(z) of degree two. Now I define the real
numbers R as the subextension fixed by this automorphism, which I now
dub "complex conjugation". Now I define an archimedian absolute value
by |z| = sqrt(z conj(z)). Now C and R are topological fields under the
topology defined by this, and conj is continuous. Now I define Q as the
intersection of all subfields of C (or R.) Z is the ring of integers of
Q. I've defined things so that, by definition, Q is a subfield of C. Of
course doing it your way I have a Q in C which is uniquely isomorphic
to the original Q which was constructed, and so forth blah blah.

And so the conclusion is that the proper answer to "is the complex number x+0i the same as the real number x" is "it depends upon your framwork."



Incidently, the problem with your way of defining the complex numbers is that it is by no means a priori obvious that this defines the complex numbers uniquely (up to isomorphism). I am going to take your word for it that it does. But I would contend that this is not a natural way to define the complex numbers.

Stephen
.

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