[FOM] MacLane

[Bas Spitters]

Dear Arnon,

On Tuesday 12 May 2009 12:29:30 Arnon Avron wrote:
>    In his Book "Mathematics: Form and function" Mac-Lane
> says somewhere that one of the main shortcomings of intuitionistic
> mathematics is that it ignores geometric intuitions (I am
> relying on my memory here, I do not have a copy of the book).

A quick search in MacLane's book gave me the following quotes:
p.404. `Thus alternative [intuitionistic] logics are closely related to 
geometry!'

Perhaps this is the quote you were looking for. 
p.443. `Brouwer's own earlier, non-formal intuitionism was based on an 
assumption of the primacy of the natural numbers in Mathematics—and this flies
in the face of the observation that actual Mathematics effectively uses
natural numbers and real numbers as equally primary. The pillarstones
are geometry and arithmetic.'
I find this quote very surprising. Brouwer posed `two acts of intuitionism'. In 
short, the first introduces the natural numbers, the second the real numbers.
E.g. http://plato.stanford.edu/entries/intuitionism
I do not see how MacLane and Brouwer disagree.

More relevant for the current discussion and related to what Andrej is saying:
p.449 This realist view [on set theory] contrasts sharply with our views on 
set theory. It seems to us that realism in sets faces great difficulties. First, 
it is open to all the objections made above to ontological and to 
epistemological Platonism. Second, that "apprehension" of sets is pretty 
obscure, since abstract sets are not apprehended by our usual senses. Third, 
there are many viable variants of the Zermelo-Fraenkel axioms, to say nothing 
of intuitionistic set theory. Fourth, because of the artificial constructions 
required this doctrine does not account well for the actual "objects" studied 
by Mathematicians. For example, holomorphic functions of a complex variable 
are central objects of Mathematics—and they are certainly not to be understood
well as "sets of ordered pairs".


Bas