[FOM] Tait on constructive mathematics

[William Tait]

Dear Mark,

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.

I should remark that I share Gabriel Stolzenberg's respect for  
Brouwer. In my case it has to do with his recognition that, between  
the Kronecker position that no objects should be introduced that are  
not finitely representable and no concepts should be introduced which  
are not algorithmically decidable and the position of classical  
mathematics that concepts built up from basic ones by means of the  
operations of logic are to be admitted and the law of excluded middle  
should apply to them, he realized that there was an intermediate  
position which merited consideration: that objects not representable  
by numbers and non-algorithmic concepts may be introduced; but one  
should simply not apply the law of excluded middle to them.

Maybe you should accuse me of inconsistency, since it was just this,  
the admission of concepts defined by means of the logical operations,  
that Weyl rejected in Brouwer's intuitionism.

This is beginning to make my head hurt.

Kind regards,

Bill

On Feb 15, 2006, at 4:36 AM, Mark van Atten wrote:

>
> On Sat Feb 11 14:41:10 EST 2006, Bill Tait wrote in `Haney and Tait on
> intuitive sources of mathematics':
>
>> Weyl, who was a better philosopher than Brouwer, understood that the
>> successor operation was not the issue, but rather that then basis of
>> arithmetic is the notion of a finite iteration of *any* operation,
>> and he took that notion of finite iteration as what is given in
>> intuition.
>
> Brouwer describes exactly this in his dissertation, where he writes of
> `the intuitively clear fact that in mathematics we can create only
> finite sequences, further by means of the clearly conceived ``and so
> on'' the order type \omega, but only consisting of equal elements'.
>
> In a footnote he explains further:
>
> `The expression ``and so on'' means the indefinite repetition of
> one and the same object or operation, even if that object or that
> operation is defined in a rather complex way'
>
> This is on p.80 of Collected Works I.
>
> Mark van Atten.
>
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