[FOM] mathematics as formal

[Steven Ericsson-Zenith]

On Mar 25, 2008, at 1:27 PM, rgheck wrote:
> I thought logic was the study of such things as validity. It's no
> convention that, if it's true that p and also true that q, then it's
> true that p and q. It may be due to convention that the word "if"  
> means
> if, etc. (Then again, it may not be.) But that's an entirely different
> matter, and it is of no concern to logic.

In my view the notion of validity applies only to convention. That is,  
the subject of validity is convention, and the question of whether the  
convention is absolute, true and complete is of interest but it is a  
separate matter.

The discussion by Hilbert and Ackermann of the Decision Problem  
[Mathematical Logic, P112] and Church's result (referenced by Hilbert)  
that denies universal validity serves, I suggest, as an adequate  
illustration that validity is a matter of conventions in logic as we  
understand it. It tells us something about those conventions. That is,  
it is presumptuous to assume, and it is not clearly shown, that our  
logical conventions are absolute, true and complete. Or rather I take  
the position that in the absence of universal validity we have yet to  
identify a logic that is not merely a pragmatic convention.

This is not a question without controversy, and Quine did take  
exception to it, wrongly in my view.

However, I will finally appeal to Rudolf Carnap's principle of  
tolerance: "It is not our business to set up prohibitions, but to  
arrive at conventions." [Logical Syntax, P51]. For without this open  
mindedness in foundational matters we can never make new discoveries.

With respect,
Steven

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