[FOM] Psychological basis of Intuitionism

sambin at math.unipd.it sambin at math.unipd.it
Mon Jun 3 10:35:40 EDT 2013

Steve, I am pleased by your post.  I have been thinking on questions  
related to the ones you put since very long ago.
I believe "since ever" that mathematics is a creation of our minds; it  
is part of the struggle of mankind to survive. The difficult question  
(to which Brouwer and other do not give a convincing answer) is to  
explain how we can reach intersubjectivity. Perhaps this is equivalent  
to your question: "just how do we do it".

My explanation is that all human abstractions, including those of  
mathematics, are built through a  dynamic process which takes place  
both in our minds and socially. So it is partly a matter of internal  
creation and partly an adaptation to what we learn from other. Here is  
where psychological analysis could come in. The key point is that  
nothing exists unless we construct it, in whatever way. In the two  
papers below you can find a more systematic exposition of this  
attitude, which I have called dynamic constructivism.

I also agree with Andrej Bauer on the non-idelogical motivations for  
constructivism in mathematics. In my opinion, the matter is only the  
quality of information that one wishes to be preserved by one's  
mathematical abstractions.


G. Sambin, Steps towards a dynamic constructivism,  in: In the scope  
of logic, methodology and philosophy of science.
Volume one of the XI International Congress of  Logic, Methodology and  
Philosophy of Science, Cracow, August 1999,
P. Gärdenfors,  J. Wolenski and K. Kijania-Placek eds.,  Synthese  
Library 315, Kluwer 2002, pages 261 - 284

G. Sambin,  Real and ideal in constructive mathematics, in:  
Epistemology versus Ontology, Essays on the Philosophy and Foundations  
of Mathematics in honour of Per Martin-Löf, eds. P. Dybjer, S.  
Lindstöm, E. Palmgren and G. Sundholm,  Logic, Epistemology and the  
Unity of Science 27, Springer  2012, pages 69 - 85

Quoting Steve Stevenson <steve at clemson.edu>:

> “Intuitionism is based on the idea that mathematics is a creation of
> the mind. The truth of a mathematical statement can only be conceived
> via a mental construction that proves it to be true, and the
> communication between mathematicians only serves as a means to create
> the same mental process in different minds.”
> (http://plato.stanford.edu/entries/intuitionism/)
> Now that I'm retired, I would like to research how exactly this plays
> out, especially metacognitive processes. The question, simply put, is
> “Just how do we do it?”
> Some mathematicians who have commented are Poincare, Hadamard, and
> Polya. Smullyan's work in refutation starts with the human aspect and
> so did Lakatos. I am also familiar with the Craik and Johnson-Laird
> mental models view of reasoning.
> I would appreciate any comments FOM readers can provide.
> --
> D. E. (Steve) Stevenson, PhD
> Emeritus Associate Professor,
> School of Computing, Clemson University
> "Those that know, do. Those that understand, teach," Aristotle.
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> FOM at cs.nyu.edu
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