[FOM] Classical logic and the mathematical practice

[Jeremy Clark]

On May 12, 2005, at 12:49 am, Arnon Avron wrote:

> On Wed, May 11, 2005 at 10:26:16AM +0200, Jeremy Clark wrote:
>>  I cannot see any other reason for favouring
>> classical over constructive mathematics.
>
> Can't you really?
>
> So let me give you two reasons.

Your reasons are, of course, the usual ones. You present them as if I 
and other constructivists are unaware of them or "completely ignore" 
them, but in reality we do something completely different: we criticise 
them. The lack of any adequate rebuttal to these criticisms in the 
literature (beyond the claims of intuition, that it just feels good to 
believe in excluded middle) leads me to believe that the classical view 
is held for reasons of tradition, as I said.

> One simple reason is that most mathematicians are platonists. They
> believe that what they talk about is meaningful, and that propositions
> about it are either true or false. And for people with such views
> classical logic is simply *valid*, beyond any doubt.

Yes, you are right: people who believe the law of excluded middle *do* 
have little doubt about the axiom "P or not P". Funny that.

If you read my posting to the end you see that I do not argue against 
Platonism. Your error is in believing that because your objects are 
real, statements about them must be true or false. You are simply 
stating that excluded middle follows from your philosophical position. 
Since this "argument" can only be defended by "intuition", your first 
reason can be lumped with your second: that It is "just obvious" that 
functions (and other mathematical objects) behave the way we (i.e. you) 
picture them behaving. A glance at history ought to show that intuition 
is no friend to the progress of mathematics. To various thinkers in the 
past it has been just obvious that zero is not a number, that numbers 
must be rational, that infinite series cannot converge to a finite sum 
... Does your intuition make room for space-filling curves? 
non-principal ultrafilters? regular cardinals? Do you *really* have 
intuitions about regular cardinals?

Let us not send FOM readers to sleep with another reiteration of 
arguments for excluded-thirdism which have already been trounced many, 
many times before by people far more articulate than myself. To see 
this trouncing in action without leaving your terminal, one need go no 
further than Fred Richman's website: Seek out "Interview with a 
Constructive Mathematician" in the preprints section. I have nothing 
original to add, but I do want to point out how absurdly misleading it 
is to have the same old arguments presented as if constructivists are 
unaware of them, choose to overlook them or have not already more than 
adequately dealt with them in the past.

Regards,

Jeremy Clark