[FOM] Psychological basis of Intuitionism

Andrej Bauer andrej.bauer at andrej.com
Tue Jun 4 09:41:53 EDT 2013

On Mon, Jun 3, 2013 at 4:35 PM, <sambin at math.unipd.it> wrote:

> The key point is that nothing exists unless we construct it, in whatever
> way.

This is a step I have never been able to follow, namely that existence of
mathematical objects depends on a mind that constructs them. I find it
awfully subjective and would in fact brand it as a form of solipsism. If
mathematics is the art of hypothetical reasoning, surely then we must
develop mathematics which is open to all possibilities, including the one
in which mathematics has an independent and objective nature, and moreover
allows for existence without construction. (This of course is not to be
confused with our ability to perceive such objectivity or our inability to
comprehend such illusive forms of existence. In any case, metaphysical
conclusions based on the limited nature of minds seem misguided.)

>From what I know of Giovanni's work, I would expect that he and I will
depart when it comes to the question whether mathematics should be
*conservative*, i.e., open to all possibilities in the sense that it only
considers the common core of all varieties of mathematics. My position is
that the conservative core is precisely that: a valuable core from which we
can grow and explore many kinds of mathematics, some of which are mutually
incompatible. The alternative, which is wrongly called "constructivism"
instead of "conservationism", dooms us to an ascetic position from which we
may only helplessly observe those who are always two steps ahead because
they do not fear thinking heretical thoughts.

Being unable to find any philosophical reasons for accepting this or that
kind of mathematics, I am luckily reduced to searching for answers *within*
mathematics. There I have found ample evidence that constructive
mathematics has its place not on the fringe, but at the core of mathematics.

With kind regards,

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