# Constructive Mathematics: Proof and Computation

## Final Report Summary - CONSTRUMATH (Constructive mathematics: proof and computation)

Although constructive proofs (that is, proofs which actually show how to construct, compute, or find the objects the existence of which is under scrutiny) have been highly regarded in mathematics for a long time, only in the last few decades significant concerted efforts have been devoted by a number of mathematicians and theoretical computer scientists to develop deep mathematics constructively. Within the frame of CONSTRUMATH, the notion of constructive mathematics specifically refers to mathematics with intuitionistic logic and some appropriate set-theoretic or type-theoretic foundation. The CONSTRUMATH project has been oriented towards shaping a more coherent and solidly based body of constructive mathematics, in the sense that classical, non-constructive mathematics can be said to be coherent or well-based.

To achieve this goal, the participants have focused on their respective areas of expertise and their interrelations, and in particular on the following topics:

(a) constructive analysis, algebra, and topology (including 'point-free' formal topology);
(b) the extraction and implementation of programmes from constructive proofs;
(c) constructive reverse mathematics.

The work performed can be classified according to five work packages (WPs):

- WP1: Constructive analysis - Further development of constructive analysis in the sense of bishop
- WP2: Algebra - Extended and deepened the use of finite methods in algebra in various directions
- WP3: Reverse mathematics - Classified various theorems in intuitionistic, constructive recursive and classical mathematics by logical principles, function-existence axioms and their combinations
- WP4: Programme extraction - Study the theoretical foundations and also address more practical problems pertinent to the area of programme extraction from proofs. Further development in the study of approximable functions in type theory and programme extraction for constructive analysis
- WP5: Topology - Further development of a minimal foundation for mathematics and consideration of its applications as well as of its philosophical implications. New results regarding apartness spaces were discovered and a systematic study of overlap algebras was undertaken
- WP6: Trimester - The training trimester took place in Munich and Uppsala in November and December 2009
WP7: Workshop - The project workshop was held in Frauenchiemsee, near Munich, in June 2010.

In addition to presenting the results of CONSTRUMATH to a wider audience, this project workshop has given major impetus to perhaps the most promising development that has taken place in mathematical logic and computer science in the past decades: that is, the undertaking to link homotopy theory and type theory which has been started by Voevodsky et al. (see http://homotopytypetheory.org/ online for further details).

The exchange programme resulted in the following:

- it enabled researchers in constructivity (both those working with constructivity in the abstract and those involved in extracting and implementing computer programmes from constructive proofs) to bring each other up to date with the most recent work in their fields;
- it facilitated existing contacts between researchers of the participating groups;
- it fostered new research collaborations between the participating groups, and in particular enabled younger researchers to get to know, and to start collaboration with each other.

The partners have been producing refereed publications in peer-reviewed, international research journals and conference proceedings. As all participating groups are amongst the leading research groups in the area of constructive mathematics, the relevant research worldwide was affected to a considerable degree.