[FOM] Tait on constructive mathematics

Mark van Atten Mark.vanAtten at univ-paris1.fr
Thu Feb 16 04:41:10 EST 2006

Dear Bill,

> My remark that Weyl was a better philosopher than Brouwer was  
> superfluous, but [I whisper] nevertheless true. Brouwer wrote many  
> things---not always very clearly----and no doubt he touched on finite  
> iteration. But when he spoke about the 'acts of intuition' at the  basis 
> of mathematics, he mentioned two-ity, which yields the  (Kantian) 
> successor operation, but he did not mention finite  iteration, which 
> must be understood to apply to all operations and  not just the 
> successor operation. In contrast, Weyl (in his  intuitionistic phase) 
> explicitly realized that finite iteration was  the basic intuition (I 
> would call it a concept) underlying the number  concept.

Brouwer is fully explicit on all this, and on the fact that these
notions all come from the basic intuition. In his dissertation, he
writes that basic notions such as `continuous', `entity', `once more',
`and so on', are `immediately conceived in the basic intuition or
intuition of the continuum' (and therefore irreducible) [CW I, p.97]. In 
a handwritten note in his own copy of his thesis, Brouwer explains that
these notions are just so many different `polarizations' of the basic
intuition [p.136 in D. van Dalen, ed., L.E.J. Brouwer en de grondslagen
van de wiskunde, Utrecht:Epsilon 2001]. Thus, each basic notion arises
from isolating a certain aspect of the basic intuition that comes to the
fore when taking a different perspective on it.

This is entirely consistent with his formulation of the First Act e.g. 
on CW 510 (the paper `Historical background...'). In fact, he doesn't 
specify any particular kind of constructions in the First Act, he just 
says that the `empty two-ity' is `the basic intuition of mathematics'.

(Compare `The only possible foundation of mathematics must be sought in 
this construction under the obligation carefully to watch which 
constructions intuition allows and which not.' [CW I, p.52])

The explanation of iteration on p.80 of CW I, therefore, is hardly an
afterthought, but a report on a further exploration of the basic 
intuition, which should be read as such, in order not to misconstrue 

Note that if one considers choosing a number an admissible operation,
one obtains the notion of choice sequence from what Brouwer says in that 
footnote; and in fact---given the above, an unsurprising fact---Brouwer 
later observed that indeed the second act of intuitionism (the 
generation of choice sequences) is just a special case of the first act. 
[p.93 of D. van Dalen, ed., Brouwer's Cambridge lectures on
intuitionism. Cambridge University Press, Cambridge, 1981.]

As ever,

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