FOM: 'constructivism' as 'minimalistic platonism'

Peter Schuster pschust at
Thu Jun 1 12:55:11 EDT 2000

I tend to view Bishop-style constructive mathematics (BISH) as based on 
some kind of 'minimalistic platonism' which presupposes the 'existence' 
of only one 'idea', namely, that of the infinite collection of natural 
numbers, just as Brouwer's intuitionism is based on the presumably less 
clear concept of the continuum. 

Consequently, one does neither have to distinguish between 'potential' 
and 'actual' infinite nor to stick to the Brouwer-Heyting-Kolmogorov 
interpretation of the logical connectives, let alone to that of 
universal-existential statements standing behind countable choice. 
Moreover, this 'platonistic' character clearly distinguishes BISH from 
any formalism in sheep's clothes related with attributes such as 
'effectively computable'. 

The reason why to choose intuitionistic logic is then the insight due 
to Brouwer that tertium non datur can hardly be applied in infinite 
contexts: who is legitimated to decide whether there is any odd perfect 
number? Note that this is rather an epistemic issue than one of subjectivism, 
let alone one of solipsism: the choice of the logic is directed by the 
assumption that there is no a priori knowledge of such matters. 

Peter Schuster

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