Well, I can’t compete with Louise’s, Tommaso’s and Mottle’s blogs for conference blogging this week, so let’s continue with a few remarks on p-adic logic. Mahndisa made an interesting comment on kneemo’s blog. She said:

Any three vertex shape could represent three adicity in context and the lines that connect each vertex could be seen as a swapping morphism of sorts. I played around with this geometry a while ago and you can extend it up to n rational dimensions. A four adic system is represented by a trapezoid, rectangle, rhombus or square, each of the vertices are connected via lines and each of these lines represent swapping.

Mahndisa has been studying Matti Pitkanen’s p-adic physics for quite some time now, and has a unique perspective on where all this is heading. From a categorical point of view, it is interesting to note that we are discussing new structures. I recall Batanin discussing a mysterious kind of operad which swaps sources and targets. Recall that sources and targets are idempotents for us.

Thus the usual 2-adic situation is the tip of the iceberg. And since categories can arise as algebras for operads, the p-adic logic would give rise to new kinds of category.

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## Mahndisa S. Rigmaiden said,

March 26, 2007 @ 10:15 pm

03 26 07

Hey Kea:

Thanks I think I will post about this. I was looking over some notes from last year and some diagrammatic reasoning really shows up, and this occured before I studied Matti’s theory. But of course, his theory augments what little I know. Thanks:)

## Kea said,

March 26, 2007 @ 11:39 pm

I look forward to your post!

## Matti Pitkanen said,

March 27, 2007 @ 11:37 am

Dear Kea and Mahndisa,

my strong belief is that one must be really ambitious and start from the condition that real physics and various p-adic physics (including algebraic extensions of p-adics and even finite-dimensional extensions such as those involving p powers of e) must be fused to single coherent whole.

This leads naturally to the generalization of the notions of the number and manifold and to the ideas that I summarized in previous posting.

By the way, I made a big step in the understanding of particle spectrum. As usual it was based on unpleasant question which always emerges after empty-head period.

My previous belief was that only Higgs boson corresponds to a (very tiny!) wormhole contact but the assumption that theory is free at parton level (TGD is almost topological QFT at parton level) forces the conclusion that also gauge bosons are wormhole contacts.

This simplifies enormously the predicted spectrum. It also allows to understand the distinction between graviton and other gauge bosons and allows now to understand the earlier speculative picture about stringy aspects of TGD at fundamental level.

The dramatic deviation from string models is that ordinary gravitons have length of order Compton wavelength of electron and stringy physics is responsible for the strong interactions of nuclei and central also for the understanding of high Tc superconductivity.

For a summary see my blog.

## Kea said,

March 27, 2007 @ 11:46 pm

my strong belief is that one must be really ambitious …Oh, don’t worry, we agree!