[FOM] Classical logic and the mathematical practice

[Jeremy Clark]

I think that the majority of working mathematicians are still using 
classical logic because there is a tradition of using it which is 
difficult to escape from. I cannot see any other reason for favouring 
classical over constructive mathematics. It is simply that if you 
decide to do constructive mathematics, you will find yourself out of 
step with the majority of practising mathematicians, so it is a 
difficult (and perhaps career-threatening) decision, to choose to 
specialise in constructive mathematics. Moreover, mathematics is 
usually taught in such a way that one has already a strong grounding in 
using classical assumptions by the time one realises that there is any 
viable alternative.

I do not think that it is necessarily a question of Platonism. 
Constructive mathematicians do not argue over the objects of 
mathematics, just over what one can meaningfully say about them. This 
is, I think, a long-standing misrepresentation of the constructive 
position. I don't think anybody does constructive mathematics because 
it is "safer", or because of some remote philosophical preference: 
rather it is generally found to be more meaningful and less 
question-begging than classical mathematics. As Fred Richman (a 
constructivist par excellence) says somewhere, we are all Platonists, 
really, when the chips are down.

By the way: if you ever do think about working seriously in 
constructive mathematics please get in touch with me, as I am currently 
doing research in constructive (Bishop-style) complex analysis, and I 
think it is an area that deserves (and rewards) more attention.

Regards,

Jeremy Clark



On May 10, 2005, at 9:43 pm, Moshe David wrote:

> Dear FOM memebrs,
>
> I'm a young researcher in complex analysis  and I have a bothersome
> question...
> If you could answer my question I would be grateful !!!
>
> Q : Why the majority of the (working) mathematicians are still using
> classical logic ? to sharpen my question :
> we use the excluded middle without any worry and say that the real 
> field is
> the disjoint union of  \Bbb Q and \Bbb Q^c
> though we know that there will be a chance that the rationality of 
> some real
> numbers (e.g. the Euler constant) is undecidable assuming ZFC.
> Is the using of classical logic is not actually Realism/Platonism ? , 
> is
> there any ontological or epistemological justification to use classical
> logic when we know
> that intuitionist logic is more safer and remote from Realism  ?
>
> Best regards,
>
> Moshe David
> Math. Dept. BIU
>
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