[FOM] mathematics as formal

[Steven Ericsson-Zenith]

On Mar 30, 2008, at 11:46 PM, Alex Blum wrote:
> It is clearly a matter of convenion what 'true' , 'and', and 'valid'
> mean but surely given what they do mean, it is necessarily true that  
> if
> p and q are both true then p is true. Or, that if p and q have both  
> been
> proved then p has  been proved.

It is not merely the meaning of terms that is convention. The rules of  
logic are conventions too.

I would not disagree "it is necessarily true that is p and q are both  
true then p is true" but, this is a feature of the convention (and,  
incidentally, vacuous as it stands). The rule appears so compelling  
because it is a property of the convention, not because it is a  
property of the world.

We should remember that these rules, these "mechanics," were the  
product of a refinement over some period that has indeed made them  
increasingly useful. But that refinement did not imbue them with  
"absoluteness," it simply imbued them with "usefulness."

> Just imagine the contrary in either
> case. And if validity is a matter of convention what of inconsistency?
> For if p is valid not-p is inconsistent.


This does not alter the argument. These remain features of the  
convention. Obviously, these happen to be useful features. Indeed,  
very useful, because I take the feature of incompleteness, for  
example, to be a feature of the convention, not of the world. Indeed,  
this was Carnap's position too, see Carnap's comments on P 907, in  
Schilpp's "The Philosophy of Rudolf Carnap":

"... we know from Godel's result that a general concept of logical  
consequence cannot be defined constructively. But this does not  
exclude the possibility, and at the present moment I see no reason for  
abandoning the hope, that a satisfactory modal logic may be  
constructed in the future in which the symbol of strict implication  
can be interpreted in an unrestricted sense, i.e., not constricted by  
reference to constructively specified rules of deduction."

To clarify, because Carnap means to say (and he does say in the  
following) "... cannot be defined constructively" ... *as we currently  
understand the convention of logical construction*. He does not mean  
that we need to abandon the general principle of logical construction,  
but simply that the particular nature of that construction must  
change. This change will lead us to a further refinement of our  
conventions that we will no doubt measure by its usefulness. But its  
absolute nature must forever be an open question. That is, after all,  
a requirement of the scientific method.

With respect,
Steven

--
Dr. Steven Ericsson-Zenith
Institute for Advanced Science & Engineering
http://iase.info
http://senses.info