[FOM] How much of math is logic?

[Andrew Boucher]

Shipman has asked "How much of math is logic?"

I'd like to plug one of my own papers which bears on this question.

In it I show that, in second-order logic with comprehension, one can  
use definitions alone to prove a sub-theory of Peano Arithmetic.   
This sub-theory is essentially equivalent to Peano Arithmetic without  
the Successor Axiom.  (Remark this sub-theory is essentially a second- 
order version of Raatikainen's Truncated Arithmetic in "The Concept  
of Truth in A Finite Universe").    The sub-theory is non-trivial, as  
it can prove e.g. Quadratic Reciprocity.

I have notes which show (so this is modulo writing it up, and errors  
may exist) that more complicated definitions allow one to dispense  
with comprehension altogether, i.e. second-order logic without  
comprehension aka two-sorted first-order logic, along with  
definitions, suffices to establish this same sub-theory.

The paper (in pdf format) is called "Who Needs (to Assume) Hume's  
Principle?" and is available here:  http://www.andrewboucher.com/ 
papers/whoneedshp.pdf