[FOM] How much of math is logic? A positivist view.

[Steven Ericsson-Zenith]

Logicism - that I take to mean the founding of mathematics upon a  
strict logical basis - remains an item on the agenda of positivism.

Frege expresses concern about the positivist requirement:

"For ultimately, the role of the infinite in arithmetic is not to be  
denied; yet, on the other hand, there is no way it can coexist with  
this [positivist] epistemological tendency." (reported in Martin  
Davis's "Engines of Logic")

Frege's comment serves to highlight where logical positivism  
inevitably draws the line - infinity is to be denied.

Rather than posing a challenge to positivism this observation  
highlights the challenge by positivism to the very notion of  
infinity. For positivism the line between logic and mathematics is  
the apprehensible. The empty set is apprehensible, the infinite set  
is not.

It follows, in this view, that recursion is logically invalid unless  
conditionally bound and repetition is invalid unless it has a finite  
constraint.

 From the positivist corner Logicism requires the elimination of  
infinities from mathematics - and the challenge of logicism is not  
whether logic can provide a foundation for mathematics but rather  
what changes must be made to mathematics to enable that foundation.

With respect,
Steven

PS. Thanks to Martin Davis for his private comments on this question  
and for bringing to my attention Frege's remark.

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