Good comments. This is interesting stuff. In extending Imaginary Logic to N valued cases, there were some unexpected but fun things I discovered in vector spaces, Number Theory, Hodge Theory, and pathologies in Topology such as the Long Line and aspects of large cardinals.
The draft paper, 50 pages long or more, languishes, as I’m up against deadlines for 12 papers I’m co-authoring for one conference along in a month or so (the 7th International Conference on Complex Systems, Boston, 28 Oct - 2 Nov 2007) and the distracting bureaucracies of public school and grad school education.

Math keeps me sane. I think. Of course, I’d think so even if I were not sane. That Zen enough?

So that’s one sane mathematician. Now to verify the others…

I’m wondering whether this James Flagg mentioned by Kauffman is “the same” (after a temporal delay! ) as someone now known as Sohaku Flagg

Well, I can ask him tomorrow morning…

There’s certainly a wealth of suggestive material, so I’ll just confine myself to a brief remark on this I = not(I) business. The sense in which the logic of Imaginaries retains its Boolean character is by implementing the so-called Flagg resolution, which can be understood in very ordinary terms, it seems to me. Namely, the free Boolean algebra B(I) on one generator I has 4 (= 2^(2^1)) elements, whose values will be denoted 0, I, J = not(I), and 1. Being free, there’s a unique Boolean algebra involution

Flagg: B(I) –> B(I)

which sends I to J (and therefore J to I). Thus, in applying Flagg to a Boolean expression in the atom I, the rule is to replace I by not(I) in every instance where it occurs, and all Boolean identities are preserved. (The way Kauffman puts it is that we are restricting the rule of substitution of one element for an “equal” element, by implementing substitution globally at every instance instead of just locally, but I think that’s really equivalent to the reformulation I’m giving here.)

The situation seems to me precisely analogous to Galois theory, where the two “imaginary” roots of x^2 = -1 are alike in every respect, and in that respect “indistinguishable”, although not actually equal [perhaps Kauffman might say they are "equal" across a time delay of applying an automorphism], so that Galois theory resolves the “paradox”.

One could also relate Kauffman’s temporal interpretation to the obvious involution e o on the Boolean topos Set/{e, o}.

A random question on trivia: I’m wondering whether this James Flagg mentioned by Kauffman is “the same” (after a temporal delay! ) as someone now known as Sohaku Flagg, who describes himself as a Rinzai Zen priest. I was in the audience at a presentation of tea ceremony by Rev. Flagg, and in his biographical remarks he mentioned he had begun working toward a Ph.D. in mathematics in the Chicago area (where Kauffman resides) before he began studying Zen in Japan. Would anyone have information on such an obscure detail?

Interesting that Kauffman and Varela were once co-authors!

Several of these references explain it better than I do,

REFERENCES:

William Bricken, “A Complex Logic for Computation with Simple Interpretations for …”, 2002
http://www.boundarymath.org/papers/CompLogic.pdf

Jeffrey James, “Interpretations of Laws of Form”,
Copyright © 2000-2004 by Richard Shoup
http://www.lawsofform.org/interpretations.html

Louis H. Kauffman , “Time, Imaginary Value, Paradox, Sign and Space”, AIP Conference Proceedings, 2002
http://www.math.uic.edu/~kauffman/TimeParadox.pdf

L.H. Kauffman and F. J. Varela, “Form Dynamics”, J.
Soc. And Biological Structures, 1984.

L.H. Kauffman, “The Robbins Problem – Compuuter Proofs and Human Proofs, Festschrift in Honor of Gordon Pask
[has star-algebra examples, but admits not knowing
what they are so named, or what they are for]

L.H. Kauffman, “Imaginary Values in Mathematical
Logic”, Proc. 17th Intl. Conf. Multiple Valued Logics,
26-28 May 1987, Boston, IEEE Comp Soc Press, 282-289.

L.H. Kauffman and H. C. Sabelli, “The Process
Equation”, Cybernetics and Systems, Vol.29, pp.
345-362, 1998.

F. W. Lawvere, “Adjointness in Foundations”,
Dialectica 23:82, 1969.

H. R. Maturana and Francisco J. Varela, “The Tree of
Knowledge – The Biological Roots of Human
Understanding”, New Science Library, 1987.

[NLL], “Nonlinear Logic (NLL) – Making Sense Out of
Logical Self- Reference”,
http://www.novatialabs.com/Man59021.pdf

C. S. Pierce, The New Elements of Mathematics, ed.
Carolyn Eisele, Vol. IV: Mathematical Philosophy,
Chapter VI: The Logical Algebra of Boole, pp. 106-115,
The Hague: Mouton, and Atlantic Highlands NJ:
Humanities Press, 1976.

Neil J. A. Sloane, Online Encyclopedia of Integer
Sequences,
http://www.research.att.com/~njas/sequences/

G. Spencer-Brown, Laws of Form, New York: Julian
Press, 1969.

Ross Street, Quantum Groups: an entrée to modern
algebra (1998). (Provides a good overview of
index-free notation)

F. J. Varela, Principles of Biological Autonomy, The
North Holland Series in General Systems Research, G.
Klir ed., Elservier North Holland, Ch. 12: Closure and
Dynamics of Forms, 1979.

the presence of a fixed point of negation in A implies that A is degenerate.

At the risk of getting myself into even deeper water , I have a way that I think will solve this in a categorical model of a Heyting algebra (though it may not be a topos; I haven’t got that far)

One of the problems I noticed is that we tend to want False to be represented by a non-zero element of the truth object, but in the normal order-theoretic set model of the Heyting algebra, we make the empty set bottom and the universe set top. This makes things work out because you can’t get an isomorphism between them.

If we start with a limited nno (l-nno), as decribed in my previous post, assume that zero is a function that internally deconstructs the numbers to the empty set (equivalent to saying throw everything away) “picked out” by the unique arrow from zero.

All other objects in the category have a unique inclusion into this l-nno, that preserves the zero arrow (base case) and succession arrow. This is equivalent to saying that I can show how to build each object inductively with a finite repeatable operation. (There is some obvious 2-category action to be investigated here)

The l-nno is terminal and isomorphic to 1 (map zero to zero and all compositions of successor to the identity for 1.

That way 1 ( isomorphic to N) become top (true) as desired, but because the empty set is embedded in each object as a base case, it can also serve as the truth object. Negation now becomes interesting, because I can always negate true (throw everything away), but to negate true I have an infinite (non-terminating) chain of construction to make, as desired. The fix-point of negation obviously exists as the limit: identity.

A candidate power object N^1 obviously exists: it amounts to a proof that I have constructed the identity.
Working out the subobject axiom (or equivalent) for this has not be done yet, but so far so good.

There is obviously a lot of work to do here to make this a real solid category (which may run aground), but I think if there is any chance of success in my project we need to diverge a wee bit from some of the usual set-oriented assumptions.

I think it’s an interesting pipe dream you have, but my guess would be that it’s rather difficult (both mathematically and in terms of convincing the skeptics, who would come in no short supply!).

Don’t get me wrong: I’m not a crank who either thinks he’s on the verge some breakthrough theory, or who underestimates how hard it would be to convince a whole field that they should give up one of their favorite beliefs about their field. I’m more setting out a philosophy of math and an informal research program for myself.

Having spent a lot of time boiling down foundations for myself, I’ve noticed that the computable approaches to math are pretty darn solid, with many different models, while the uncomputable bits are a bit wobbly.

Many other people don’t seem to be troubled by these nor have thought a lot about them it seems; they seem to accept them by tradition. If I accomplish nothing else than inspire a couple people to kick at the table legs one more time to convince themselves that they really believe that stuff, or confirm what they really know about it, I’m probably still ahead.

I think I now understand the claim that a topos is not the setting for a constructive set theory.

Well, there is clearly a lot more to say on that score, but let’s save that for another day (my understanding of what is usually meant by constructive set theory is pretty sketchy, and I’m confused about some things here that I’d prefer to mull over first).

Is your contention that this is a lost cause, i.e. either put up with uncountables or lose analysis as anything more than a clever hack?

I wouldn’t claim that much just yet; I was just illustrating some things one should keep an eye on. I think it’s an interesting pipe dream you have, but my guess would be that it’s rather difficult (both mathematically and in terms of convincing the skeptics, who would come in no short supply!).

You can now recheck my calculation that you quoted at the top, that for any internal Heyting algebra A (e.g. the subobject classifier in any topos), the presence of a fixed point of negation in A implies that A is degenerate.

I think I now understand the claim that a topos is not the setting for a constructive set theory.

Finally, let me point out that while a data type N* (N plus infinity) could be an interesting surrogate for the ordinary N

I’m not proposing augmenting N in any way, but a limit for the compositions of the arrow: it would converge to the identity on N, which of course is given. (This is in fact a domain theoretic idea I’m working with.)

However I’m not THAT surprised that categories that have been currently defined don’t do the thing I’m looking for: I’m not sure how much people have looked for such a beast.

Is your contention that this is a lost cause, i.e. either put up with uncountables or lose analysis as anything more than a clever hack?