In this paper <http://www.sciencedirect.com/science/article/pii/0304397584901117> page $19$ or $29$ it seems to imply there is a difference between length of proof and length of its presentation in Peano Arithmetic.


What exactly does this mean?



What is difference between length of proof and length of its presentation in Peano Arithmetic?
Very roughly speaking: 


Suppose I deduce Theorem A from Propositions B, C, and D. 


The "length of proof" of Theorem A is 



> 
> Length of my proof of Theorem A + length of proof of Proposition B + length of proof of Proposition C + length of Proposition D + lengths of statements of Propositions B, C, and D
> 
> 
> 


The "length of the presentation" is 



> 
> Lengths of statements of B, C, and D + MAX(length of proof of A, length of proof of B, length of proof of C, length of proof of D). 
> 
> 
> 




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To put it in a more physical context: 


Length of the proof of the "fundamental theorem of calculus" is about 7 or 8 chapters of text, according to most college textbooks. 


Length of the presentation of the "fundamental theorem of calculus" is about 3 blackboards (going by most college classroom set-ups). 




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See p29 of the article you linked to, where the concept of "presentation of proof" is defined (in particular you can "erase" things you don't need later. 


