Looking From The Inside And From The Outside

  title={Looking From The Inside And From The Outside},
  author={Alessandra Carbone and S. Semmes},
Many times in mathematics there is a natural dichotomy betweendescribing some object from the inside and from the outside. Imaginealgebraic varieties for instance; they can be described from theoutside as solution sets of polynomial equations, but one can also tryto understand how it is for actual points to move around inside them,perhaps to parameterize them in some way. The concept of formalproofs has the interesting feature that it provides opportunities forboth perspectives. The inner… 
Turning cycles into spirals
In [ 131 Parikh proved the first mathematical result about concrete consistency of contradictory theories. In [6] it is shown that the bounds of concrete consistency given by Parikh are optimal. This
Pathways of deduction
This paper would like to show how the usual hierarchical approach to the construction of formal mathematical proofs is inappropriate to reveal the intricate structures underlying proofs.
Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic (Preliminary Draft)
This article is the first of an intended series of works on the model theory of Ultrafinitism. It is roughly divided into two parts. The first one addresses some of the issues related to
Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetics
A model of ultrafinitistic arithmetics based on the notion of fuzzy initial segments of the standard natural numbers series is presented, through which feasibly consistent theories can be treated on the same footing as their classically consistent counterparts.
N ov 2 00 6 Model Theory of Ultra nitism I : Fuzzy Initial Segments of Arithmetic ( Preliminary Draft )
This article is the rst of an intended series of works on the model theory of Ultra nitism. It is roughly divided into two parts. The rst one addresses some of the issues related to ultra nitistic
Streams and strings in formal proofs
  • A. Carbone
  • Computer Science, Mathematics
    Theor. Comput. Sci.
  • 2002
Where the Buffalo Roam: Infinite Processes and Infinite Complexity
These informal notes, initially prepared a few years ago, look at various questions related to infinite processes in several parts of mathematics, with emphasis on examples.


Some Combinatorics behind Proofs
We try to bring to light some combinatorial structure underlying formal proofs in logic. We do this through the study of the Craig Interpolation Theorem which is properly a statement about the
Cycling in proofs and feasibility
There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an
Existence and Feasibility in Arithmetic
  • R. Parikh
  • Mathematics, Computer Science
    J. Symb. Log.
  • 1971
“From two integers k, l one passes immediately to k l ; this process leads in a few steps to numbers which are far larger than any occurring in experience, e.g., 67 (257729) . Intuitionism, like
The first systematic account of the theory of elliptic functions and the state of the art around the turn of the century. Preceding general class field theory and therefore incomplete. Contains a
Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination
This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be,
The Relative Efficiency of Propositional Proof Systems
All standard Hilbert type systems and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes, and extended Frege systems are introduced, which allow introduction of abbreviations for formulas.
Kolmogorov Complexity and its Applications
  • Ming Li, P. Vitányi
  • Computer Science, Mathematics
    Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity
  • 1990
Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory and Proof Theory
  • W. Craig
  • Mathematics, Computer Science
    J. Symb. Log.
  • 1957
The Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms, showing that the expressive power of each first-order system is rounded out, or the system is functionally complete.
Interpolants , cut elimination and flow graphs for the propositional calculus
We analyse the structure of propositional proofs in the sequent calculus focusing on the wellknown procedures of Interpolation and Cut Elimination. We are motivated in part by the desire to
Geometry of interaction III: accommodating the additives
The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a