Topic  People  Description 

Hilbert's Programme for Abstract Mathematics 
Peter Michael Schuster

Extracting the computational content of classical proofs in conceptual mathematics. Particular attention is paid to invocations of logical completeness in mathematical form, typically as variants of Zorn's Lemma. 
Philosophical aspects of logic and foundations 
Ruggero Ferro

Several results in mathematical logic point out and explain the limitations, possibilities and advantages of formalization (the use of formal languages). An increasing precision in determining of the role of formal languages is basic to a critical attitude in philosophy of mathematics, spotting untenable positions and supporting others. An empiricist point of view is being developed that overcomes the vagueness and difficulties of know presentations. This type of research has developed, and will continue to support, competences on the following themes: Mathematical logic; Understanding, acquiring, and constructing basic mathematical notion, in particular the primitive ones; The role of logic in the construction and acquisition of mathematical notions; The role of the language in mathematics; The role of formalism in mathematics; Mathematics teacher’s initial and life long education; Mathematical motivations for the teaching of mathematics. 
Proof theory and constructive mathematics 
Peter Michael Schuster

Proof theory at large studies mathematical proofs, which thus become themselves objects of mathematics. In a nutshell, the goal is to understand "what can be proved with what" and to gain computational information from proofs. Constructive mathematics aims at direct proofs from which one can read off algorithms; any such algorithm comes with a certificate of correctness for free, which just is the original proof. 
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