At 9:36 -0700 1/10/2002, Tim May wrote:
>MWI looks, then, like just another variant of "modal realism." To
>wit, there IS a universe in which unicorns exist, and another in
>which Germany won the Second World War, but these universes are
>forever and completely out of touch with us.
Not quite due to possible interferences. We do have empirical evidences
for those "worlds" imo. (if only the two slits + Bell or better GHZ)
>>BTW, Tim, I am discovering n-categories. Quite interesting. John Baez
>>has written good papers on that, like his categorification paper.
>>Have you read those stuff. Could be useful for the search of coherence
>>condition in "many world/observer" realities ...
>I've been reading Baez for a while. An excellent teacher. I hear
>he's working on a book on n-categories. And Baez and my namesake, J.
>Peter May--unrelated to me, are leading a consortium to research
>n-categories more deeply. I confess that I have only vague ideas
>what they are....sort of generalizations of natural transformations,
A very natural generalisation (!). Just replace the hom Sets by hom Categories.
In which you can again replace the hom sets by hom categories ....
What is intriguing is the existence of coherence conditions making those
constructions apparently very genuine for many stuff from quantum
>(I'm still studying categories at a more basic level, having "jumped
>ahead" to other areas, as is my wont.)
>His "From Categories to Feynman Diagrams" (co-authored with James
>Dolan) and several of his related papers are good introductions.
Thanks. I didn't see this one. Very nice. http://arxiv.org/abs/math.QA/0004133
>Chris Isham is also very good on drawing the connections between
>conventional quantum mechanics (i.e., stuff in the lab, not
>necessarily quantum gravity or quantum cosmology) and category/topos
>theory. (In particular, the collapse of the wave function and
>measurement looks like a subobject classifier, or, put another way,
>the usual transition from "neither true nor false" in a Heyting
>algebra to the "one or the other" we _always_ see once there is any
>chance to observe/measure/decide. That is, Heyting --> Boolean is
>what the mystery of QM centers around.
Boolean or Heyting Toposes or cartesian closed categories have exponentials.
They describes subjects (first or plural). They are distributive categories.
You always seem to forget the non distributive categories (nicely introduced in
Lawvere Shanuel book), which are akin to linear and quantum logic,
and quantum algebra.
Technically linearity seems to be a consequence of the non distributivity,
I'm not sure I really grasp the idea yet.
>(I am intrigued to find that Jeffrey Bub, in his "Interpreting the
>Quantum World," 1997, makes central use of possible worlds,
>lattices, and such. While he does not explicitly mention Heyting
>algebras, the connection is close, and is implicit in the math. Had
>I encountered this approach when I was studying QM, I might have
>pursued it as a career. Instead, I was bored out of my mind solving
>partial differential equations for wave functions inside boxes. Ugh.)
>I'm reading Graham Priest's "An Introduction to Non-Classical
>Logic," 2001, which covers various modal logics, conditional logics,
>intuitionist logic, many-valued logics, and more ("first degree
>entailment," "relevant logic," etc.).
I should read it. One day I will make a comment about its use of Godel in his
book "In contradiction".
>The tableaux approach is new to me. They look like the trees of
>Smullyan, and hence like semilattices.
I have used the smullyan trees for the G and Co. theorem provers. The tableaux
structure reflects in some way the Kripke structure. Posets appears with
S4-like modal logic.
You should study Gentzen presentation of logic which are naturally related
to categories. An indigest but brilliant introduction to many (intuitionnist)
logics is the North-Holland logic book by Szabo: Algebra of proofs.
To bad he miss the braided monoidal categories ... For a categorician, knots
theory is a branch of logic.
>(I'm also reading Davey and Priestley's "Introduction to Lattices
>and Order," along with parts of Birkhoff's classic, and the
>lattice/poset approach continues to appeal to me greatly.
Nice. They have chapters on the non distributive order structures.
>It's a vantage point which makes all of this heretofore-boring-to-me
>logic stuff look terribly interesting. I'm viewing most
>programs/trees/refinements/tableaux as branching worlds, as possible
>worlds (a la Kripke), to be further branched or discarded.
>Hence my focus on MWI and "Everything" remains more on the
>mathematics. (I just ordered my own copy of Goldblatt's "Mathematics
>"Possible worlds," something I only encountered in any form (besides
>Borges, Everett, parallel universes sorts of references) in the past
>several years, is my real touchstone.
>And, more mundanely, I think it applies to cryptography and money. I
>had a meeting/party at my house a few weeks ago with about 50 people
>in attendance (gulp!). We had a series of very short presentations.
>I gave a very rushed 10-minute introduction to intuitionistic logic,
>mainly focused on my "time as a poset, a lattice" example, citing
>the natural way in which "not-not A" is not necessarily the same as
>A. If the past of an event is A, then not-A is its future. But the
>not-future is larger than the original past, as "incomparable" (in
>the poset/trichotomy sense) events influence the future. Or, put in
>relatitivity/cosmology terms, which many people are more familiar
>with, ironically, events outside the light cone of the present
>figure into the future. So the natural causal structure of spacetime
>is intuitionistic, a Brouwerian lattice.
Very plausible. But be careful of the solipsist move here.
Unless I miss something, like a universal first person may be, I really
I think---with comp---
that those brouwerian lattices emerge from the non distributive structures
which rises from the coherent glueing of all little pieces of consistent (in a
logical sense) histories.
>Anyway, I managed to spend a couple of minutes relating this to the
>world of finite knowledge about such things as "money." This is
>related to belief, trust, reputation, and suchlike.
>Afterwards, a senior member of a leading crypto company came up to
>me and said he'd done work in mathematical logic in school. He said
>he's been waiting for ten years for crypto to turn into mathematical
>logic, that the focus on number theory has been a kind of diversion.
>Enough for this digression. But MWI, belief, possible worlds,
>alternate forms of logic, knowledge, category theory, toposes, and
>more are all deeply "intertwingled," as Ted Nelson would say. It's
>all math. Good stuff.
Yes. And the intertwingling grows up the deeper is the fundamental
question, isn't it?