Seminario: Analysis of Three Different Flavours. A story of examples, counterexamples, and viewpoints

Giovedì 8 Maggio 2014, ore 15:30 - Sala Riunioni VII Piano - Maarten McKubre-Jordens


Giovedì 8 Maggio 2014 alle ore 15:30 in Sala Riunioni al VII Piano della Torre Archimede, Maarten McKubre-Jordens (Department of Mathematics and Statistics, University of Canterbury, New Zealand) terrà un seminario dal titolo "Analysis of Three Different Flavours. A story of examples, counterexamples, and viewpoints".

In this talk, which surveys at least two viewpoints both about and within mathematics, work is presented in three areas: Constructive partial differential equations ("applied" constructive mathematics); Reverse constructive mathematics; Paraconsistent mathematics.
A (very) brief introduction to Bishop-style constructive mathematics will be given. In the first topic, we provide sufficient conditions for constructive existence of solutions to Dirichlet-type problems, and outline where the classical existence proofs fail to be constructive. A corollary will be that there cannot be a universal algorithm that computes solutions to the Navier-Stokes equations of fluid flow.
In the second topic, we show a direct proof of a known result: that compactness properties (of a certain kind) follow from a non-constructive omniscience principle. The direct proof highlights the decision procedure that has to be made for those compactness properties.
The third topic ventures into the little-known realm of mathematics where contradictions may be allowed (yet sensible conclusions can still be drawn)---and surprisingly find that proofs can be very constructive in that setting.