### Klein 2-Geometry V

#### Posted by David Corfield

I had hoped to mark my first appearance in the Café with a striking new contribution to our Klein 2-geometry project. The project began on my old blog back in May, and you can follow it through its twists and turns over the next 3 monthly instalments. I have enjoyed both participating in a mathematical dialogue and, as a philosopher, thinking about what such participation has to do with a theory of enquiry. The obvious comparison for me is with the fictional dialogue *Proofs and Refutations* written by the philosopher Imre Lakatos in the early 1960s. The clearest difference between these two dialogues is that Lakatos takes the engine of conceptual development to be a process of

conjectured result (perhaps imprecisely worded) - proposed (sketched) proof - suggested counterexample - analysis of proof for hidden assumptions - revised definitions, conjecture, and improved proof,

whereas John, I and other contributors look largely to other considerations to get the concepts ‘right’. For instance, it is clear that one cannot get very far without a heavy dose of analogical reasoning, something Lakatos ought to have learned more about from Polya, both in person and through his books.

So where’s the next great step on the 2-Kleinian project, I hear you ask impatiently. Well, I have to confess that the Aquitainian sunshine, the etangs, the aroma of ripe plums, and, no doubt most significantly, the demands of my children, were not conducive to mathematical research, and so I come here largely empty-handed. I was thinking, however, that it is key to understand how vector 2-subspaces work, e.g., how to sum them, what are complements, etc.

We seemed to have established that 2-vector spaces could be characterised by 2 natural numbers *b*_{0} and *b*_{1}, its Betti numbers. All indications were that a sub-2-vector space would have to have first Betti no greater than *b*_{1}. Several indications suggested the same holds for the zeroth Betti number.

Two thoughts, then. First, if we require sub 2-vector spaces to have complements, and we knew what happens to Betti numbers when 2-vector spaces are added, this would set us the right way concerning the issue in the previous paragraph. Second, as a sub 2-group we should see what happens when we perform our quotienting operation of a sub-2-vector space acting on the 2-vector space.

Did we ever decide that we must write - sub 2-vector space, rather than 2-vector subspace? Not to speak of pushing the ‘2’ in front of the ‘space’.

## Re: Klein 2-Geometry V

[see improved version below]