Re: Don't get Axiom of Choice?

Keith Ramsey writes

On Sep 27, 8:58 am, LauLuna <laureanol...@xxxxxxxx>
wrote:
|Constructivists/definitionists require that a
mathematical object be
|constructed/defined if we are to admit its
existence.

That is true but perhaps a bit misleading. Deciding
what should be "admitted as existing" before one has
decided what it means for something to exist is to
put the cart before the horse. Many constructivists
have an understanding of the meaning of mathematical
statements, that the meaning is given by what is
required to validate the statement. As a special
case,
the meaning of an existence statement ("there exists
an x of type T such that...") is given by what is
required to validate it, which is then taken to be
what a construction of that type of thing is.

One can apply this point of view to any kind of
mathematics, assuming it has a coherent meaning.
Constructivists do then take it as being more
natural to deal with types of object where the
construction required is straightforward, for
example generating an integer in decimal form,
or supplying a method for generating it. But
this is in principle a choice. If you want to
deal with the things that are constructed by
supplying several integers in decimal form, and
associating to each one a condition, with the
property that the conditions are mutually
exclusive and it's impossible for all the
conditions to be false... then you are free to
do so. So "1 if the Riemann hypothesis is true,
2 if it is false" would be such a construction.

Constructivism does not require you not to make
such constructions; it just understands them to
be more complicated, less natural, and less
meaningful than the construction of an integer
by giving a method for computing it. Classical
mathematics reasons that these are necessarily
the same type of thing as integers, just that
we don't necessarily know how to find out which
of the possibilities has been realized.

Keith Ramsay

An excellent explanation. I think that this precept--
meaning precedes construction--is one of the most
difficult subtleties to grasp in the constructivist
philosophy.

Terms of existence are just as important to a mathematical
claim as to any claim made in natural language.

Tom
.

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