Ahoy Melbourne!
Continuing my trend of hiding from my home university, I am now at the Department of Philosophy at Melbourne Uni, visiting fellow blogician Greg Restall. Who knows, this lot may be able to transform me from a philosophical australopithecine into a philosophical neanderthal? At some point in the near future, I shall preach the gospel of higher dimensional
posted by Jon @ 9:24 AM
4 Comments:
Preaching to the converted is also fun.
Have you had a look at Todd Trimble and Geraldine Brady's work reformulating Peirce's existential graphs in terms of higher-dimensional algebra?
As I mention on my blog, Todd said to me:
"One thing I would have emphasized, had I addressed your group, is Lawvere's revolutionary insight that the connectives and quantifiers in logic are controlled by *adjoint functors*. I think this is the key to further progress in geometrizing logic: higher-dimensional adjunctions are intimately connected with Morse theory, esp. the calculus of cancelling and rearranging critical points of Morse functions. (I don't think Gerry Brady and I fully connected the Beta graphs with this geometric aspect of adjunctions -- it ought to be done.) It is interesting to me that Peirce perceived, at a pre-formal level, the structure of connectives via adjunctions."
What do you think?:
"...Summary
If logic is regarded as a set of perceptual operations, then logic has a chronotopology (time structure). Identity or nonidentity then results as a decision from an algorithm - a set of perceptual operations and comparisons - in which case the nature of a particular identity is conditional upon the nature of the set of perceptual operations comprising the algorithm.
Ordinary logic does not account for the temporal aspects of perception, merely accounting for the spatial aspects.
In other words, Aristotlean logic is a synthesis of primitive observation, fitted to the partial (spatial) reality emerging from spacetime after the imposition of the monocular (one-at-a-time) photon interaction with matter.
In quantum mechanics, time is a parameter, not an observable. Hence measurement/detection (of observables) deals with primitive observation and Aristotlean logic (topology).
Total reality includes nonprimitive observation - hence, non-Aristotlean logic (chronotopology)"
"...By applying temporal accounting to each perceptual operation, Aristotle's three laws can be shown to be self-contradictory and incomplete as written. That is, they are topolological, not chronotopological.
A simple derivation of a fourth law is shown and an application rule given which itself may be regarded as a fifth law of logic.
A proof of the fourth law by demonstration is given.
The resulting four-law logic is chronotopological. The application rule states that either Aristotle's three laws apply explicitly and the fourth law is implicit, or the fourth law applies explicitly and Aristotle's three laws are implicit.
The four-law chronotopological logic is theoretically capable of resolving every present three-law paradox."
I AM NO LOGICIAN BUT MERELY A BEER DRINKING, plebian. So please excuse my "brute force" way of posting.
Here is the link:
http://www.cheniere.org/books/aids/appendixIII.htm
David: Thanks for the pointer to the paper. It looks interesting, but I have not had a chance to read it yet.
Anon: You may be interested in reading some stuff on temporal and tense logic. In particular, if p is true today but not p is true tomorrow, is this really a contradiction, since they are true at different times? For instance, the statement "it is raining" is true on some days but not true on others. Temporal/tense logic is a formalism that allows one to incorporate such temporal information into the logical basis. See here for more details on this. For some pointers to incorporating both spatial and temporal information into a logic, have a look at this old post for a discussion of some nifty possibilities.
Well, I should wish him luck with the conversion thing. It won't be such an easy task but if he could do well on it. I'll have to bow down, on my knees.
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