[FOM] Psychological basis of Intuitionism

[sambin at math.unipd.it]

@ 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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