[FOM] Psychological basis of Intuitionism
sambin at math.unipd.it
sambin at math.unipd.it
Thu Jun 6 16:05:34 EDT 2013
@ Steve
Since you used the word "intuitionism" in the title of a post in FOM,
I understood that you meant intuitionism in the sense of FOM, that is,
the foundational school started by L.E.J.Brouwer at the beginning of
last century. After you spoke about "psychological mechanisms", it
seems to me that a better title for your questions would have been:
"Psychological basis of intuition". On this topic I have little to
say. Except for the remark that non-euclidean geometries started to
be really accepted when somebody (Beltrami, Riemann) built models of
them.
I am a declared intuitionist since over 30 years. My psychological
reasons are that I feel deeply uneasy with anything I "must" assume
without experiencing it in some way.
@ Andrej
About solipsism: If you read the two papers of mine mentioned in my
previous post, you will see that the main distinction of my view from
Brouwer's is precisely that I consider social communication as
essential for the construction of any abstraction (this was only
hinted in my first post). So it is the opposite of solipsism.
In the same time, however, I believe math should be something more
than social hallucination. That would be really solipsism, although
collective. In the second paper I mentioned this is made clear by
insisting on the fact that mathematics is made of abstraction *and* of
application.
About my minimalist approach: I like it because it shows how much you
can do with so little assumptions. Moreover, minimalism in assumptions
goes together with maximalism in conceptual distinctions. Again for
the purpose of communication, I believe it is convenient to build a
foundation which is compatible with as many other foundational
attitudes as possible. This btw proves that I am ready to accept other
attitudes.
Using my minimalist approach in an actual development of topology, new
mathematical structures have emerged which were hidden under stronger
assumptions. This I believe is at least 7 years ahead of times (as it
happened with other novelties of mine), not just 2 steps.
About independent and objective nature of mathematics: Andrej, all of
us are free to believe what we prefer. This is a fact, and it's not up
to me! But if you like to believe, as most mathematicians and
philosophers of mathematics do, that mathematics has an "independent
and objective nature" as you say, then in my opinion you should
explain where this comes from, on which basis we can say this, or
other equivalent questions. Since nobody has been able to give proper
answers, such a belief does not seem to me a foundation, but just
begging the question (of what a foundation of mathematics is). I
myself find this assumption so strong, and unfounded, that I feel very
uneasy when I meet it. That is why I prefer the minimalist approach. I
feel better from that perspective, since I can give a rationale for
the various foundational assumptions.
@ Vladik
It is not nice to charge a colleague of being dogmatic and old
fashioned before reading the papers he suggests. Understanding your
remarks as a question, my answer is that it all depends on what one
means by construction. My version of constructivism is not scholastic.
In my opinion, construction should not be limited to what is reducible
to Turing computability (which indeed would be dogmatic).
Best regards to all
Giovanni
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