There is a discussion at the Everything Seminar about everyone’s favorite topic, the axiom of choice. The axiom of choice has various pathological consequences, such as the Banach-Tarski paradox and the existence of a non-Lebesgue-measurable set. The problem, everyone notes, is that by its very nature constructions that depend on the axiom of choice cannot be given explicitly. It’s tempting to dismiss the axiom of choice on those grounds as illegitimate (as Lebesgue himself did).
This philosophy can be taken further. What if we restrict ourselves to sets we can construct explicitly? One answer leads to constructivism, which entails rejecting many other basic principles of mathematics, such as the law of excluded middle, and has a dramatic impact on how every branch of mathematics is developed. A less dramatic course of action was proposed by Gödel: the axiom of constructibility. Gödel analyzed the types of sets that you can construct from a given collection of sets in terms of nine explicitly-given functions (these are familiar operations such as taking the direct product of two sets, or taking the image of a function between two sets). Gödel then defined a universe of sets in stages. At each stage, the sets defined by repeatedly applying the operations to the sets from the previous stage. He then iterated the construction of stages (allowing the stages to indexed by ordinals) to build the constructible universe. Any explicitly given set will appear somewhere in the constructible universe. The axiom of constructibility is that every set is in the constructible universe.
What happens to the axiom of choice under the axiom? It becomes a theorem. Even more amazingly, you can define a single well-ordering over the entire universe of sets. Choice functions can be found using the well-ordering. You can even define the well-ordering reasonably explicitly: at each stage, you define each new set in terms of a finite number of the nine operations on the old sets, so to order the sets at each stage you just need to define an order on the operations. You extend this to the whole universe by defining sets that first appear at later stages to be bigger than sets from earlier stages. To find an example of a non-measurable set, or the paradoxical decomposition in the Banach-Tarski paradox, you can just take the first one under the well-ordering.
Okay, I’ll ask the obvious dumb question: how come constructive mathematics denies the axiom o f choice, while the axiom of constructibility makes it a theorem? I suspect that “constructible” doesn’t at all mean “constructive”.
John,
Axiom of constructibility, doesn’t deny AC, it instead allows every choice to be explicit, by imposing additional structure on every set (or, equivalently, by limiting universe of sets).
John, I think the cognitive dissonance is that you have to change what you mean by “set” and “function”.
Let’s write AC as “every surjection has a section”. Now the AC in our new context reads “every constructable surjection (between constructable sets) has a constructable section”. So no, the AC as we usually interpret it is not a theorem in the new context, but an appropriately modified version of it is.
My take (or guess):
The axiom of constructibility states that every object in the universe can be explicitly defined (in some sense).
Constructive mathematics requires that everything that you do with the objects in the universe must be explicitly defined. It does notallow you to assume that those objects themselves have constructions. Even if you explicitly construct a set (by giving an algorithm for determining whether an arbitrary object is a member) you don’t know that its elements themselves have definitions, so you have no way of well-ordering them.
It’s a bit like the difference between “everyone must have a valid birth certificate” and “a copy of every contract you sign must be saved in the archives” (but not very much, I wish I could think of a better analogy).
Constructible and constructive are not totally unrelated, though a genuine constructivist would never accept the higher reaches of the constructible universe. Constructible allows you to perform inductions on arbitrary ordinals, while constructive would restrict you to ordinals that are explicitly computable. An example of where the difference would matter is the construction of the Borel hierarchy, which requires induction up to the first uncountable ordinal.
As Walt says, induction on arbitrary ordinals is the difference. Constructivists don’t acknowledge the existence of non-recursively-definable well-orderings (or something like that), which means there are only countably many countable ordinals. The constructible hierarchy takes all the ordinals from the full universe, and uses them to define things.
All the hoopla about the axiom of choice always frustrates me, because in many ways it is not choice that is the problem but the acceptance of uncountable sets.
Inductively (or equivalently recursively) defined (i.e. countable) are well-behaved, and by definition are well-ordered, so generally countable sets don’t produce that nasty AC paradoxes.
You only get into trouble when you pretend that if you had an infinite amount of time you could build an infinite number of countable infinities.
I realize that rejecting uncountables requires people to relinquish their cherished belief in the reality of the continuum, but if you consider the algebraic numbers (which are countable) augmented by a few (or even a countably infinite number of) transcendental numbers of interest ( eg. e and pi ), you can still do pretty much everything you might want without having to lose sleep over paradoxes.
If you think about it, that’s what we do in practice anyway.
I realize that rejecting uncountables requires people to relinquish their cherished belief in the reality of the continuum, but if you consider the algebraic numbers (which are countable) augmented by a few (or even a countably infinite number of) transcendental numbers of interest ( eg. e and pi ), you can still do pretty much everything you might want without having to lose sleep over paradoxes.
If you think about it, that’s what we do in practice anyway.
In algebra, this is arguably true. In analysis, though, I would say that it is really convenient to have a complete, locally compact number field to work in; while I am sure that there are substitutes for these notions in constructive mathematics, it is not immediately obvious that they will be equally convenient.
I guess you have identified my bias; I tend to feel that analysis is a very useful fiction or as we say in a computational contexts a convenient “hack” that, as it is normally thought about, really doesn’t make sense from a strict foundational basis.
There are less intuitive but sound approaches (non-standard analysis perhaps) that can make things work out in the end, so I’m not that worried about it; analysis works and if pretending the continuum is real makes it easier to do, then so be it.
But then it is irrational to get worked up because it introduces paradoxes in some pathological edge cases.
This is the same thing we do every day when we use 3.14 as an approximation of pi to, say, measure how big the wheels on my nephew’s go-kart should be. It would be nuts to stay up nights afterwards worrying that the size wasn’t 100% accurate, or that if his go-kart was infinitely large it would travel backwards in time and create a time-travel paradox.
Understanding when you are approximating and when you are on solid and accurate foundational ground is the basis of logical sophistication, and I would argue that a lot of grief can be avoided by striving for it.
Let me see if I understand this correctly. Let V be the abelian group of functions from Z (the integers) to Z/2 and let U be the subgroup consisting of those functions such that the preimage of 1 is finite. Personally, I have no intuition as to whether the surjection from V to V/U has a section. But Walt tells me that, if I believe every set is constructible, then I will get an explicit section of this map.
Why is this? I think the answer is that, from the perspective of a nonconstructabilist, there are many elements in V which the constructabilist doesn’t recognize. Essentially, for a constructabilist, a “random” sequence of 0’s and 1’s isn’t a genuine function. So it is easier for a constructabilist to define a section of this map, because, for a constructabilist, both V and V/U are much smaller than a nonconstructabilist thinks they are.
Is this a good intuition?
Thinking about this stuff kept me up, so I want to take another run at it from the opposite direction.
Let’s define a new “continuum”, R minus, that admits all algebraic numbers and a countably infinite number of transcendental numbers. (This would of course, as the sum of a finite number for countable sets, be a countable set.)
Within this new augmented algebraic field, R minus, let’s define Cauchy sequences whose sequential elements are taken from it.
If we can prove that there are only a countably infinite set of distinct R minus Cauchy sequences, then it seems to me that we should have a complete, locally compact set that has all the properties we want for analysis.
Other than impredicative proofs such as Cantor’s, is there any reason to think that this couldn’t be done? Or that other desirable properties would be lost?
The convergent “well-behaved infinity” property of Cauchy sequences gives me the intuition that this might be doable, though possibly very hard.
If you substitute “constructivist” for “constructibilist” in that sentence, I think you have a valid intuition.
On the other hand, Gödel’s constructible universe is in fact larger than the normal ZF, since AC is admissible in it (i.e. it can proved as a theorem) In a sense, the constructible universe allows more paradoxes; they are virtually growing on the trees, as Walt suggests.
David, that’s basically right, except here the “constructibilist” allows all constructions performed in ordinary mathematics. Take any construction in any math book, and it’s in the constructible universe. The “non-constructibilist” believes in the potential existence of exotic objects only familiar to set theorists, like various elementary embeddings of parts of the set theoretical universe into others, or zero sharp.
Marc, the constructible universe is larger in that it proves more theorems, but it’s smaller in that any theory of sets that strictly contains ZFC will contain a copy of the constructible universe.
I suppose this sums up what is wrong with classical foundations: there shouldn’t be any more distinct objects in an axiomatic system than there are proofs, i.e. theorems.
The traditional mathematical burden of proof seems to be reversed: to rule something out, however outlandish, you need to show that it produces a contradiction.
I really believe that the quality of mathematics will improve if mainstream mathematicians adopt the more skeptical position, and take constructivist issues more seriously.
I don’t think what you’re saying is true even constructively. Take Peano arithmetic, versus Peano arithmetic minus the induction principle. The first isn’t any bigger, but it proves more theorems.
It depends on what you take the objects in the system to be. If you include the relationships between the numbers as objects in the system (analogously to the Set case, where relations and structures between sets are sets themselves). then there can’t be more objects defined by the system than theorems.
If you only want to build the natural numbers, you only need two of the axioms; most of the meat actually defines relationships between the natural numbers as “objects” of the system.
This is somewhat moot though, since if we restrict ourselves to set theories, all theorems boil down to asserting the existence of some set by means of a proof. If you say that there are is a set theory that “contains” more sets than theorems, you are saying that it allows sets to exist that can’t be proved (finitely) from the axioms. I find that problematic.
Also I just noticed a possible source of miscommunication between us. My original statement to David was about the constructible universe versus ZF (no C) and your statement was relative to ZFC (C included). My apologies for missing that when I responded.
It was unclear which system David was considering “nonconstructibilist”, so either of us might have answered his question correctly.
Marc, I don’t understand your argument at all. Peano arithmetic without induction (here I mean Robinson arithmetic) can’t even prove that addition is commutative. What extra objects does Peano arithmetic have?
Thank you for the replies. For the record, I was assuming a nonconstructabilist believed in Choice, but I see how that was unclear.
A question though, regarding the statement that there can not be more objects than theorems. A naive interpretation of this is completely false: any theory written with a finite alphabet has only countably many theorems but I assume that you believe that there are theories with uncountable models. Is your position that an object O only truly exists if there is a theorem of the form
“There is a unique X such that…”
such that this theorem is true of O?
Actually, if you believe this exact formulation there are all sorts of problems if your model has a symmetry. For example, under this formulation, you wouldn’t believe that either of the two square roots of -1 exists. But do you mean something like this?
One way to model the difference is to say that the Robinson arithmetic allows you to build a bunch of equivalence relations that are preserved under succession. You start with the identity relation (you can build it using well-founded recursion down to zero from any finite tuple) As I prove an expression of the form a+b=b+a for some fixed a and b, I’m asserting that I can build and/or find a succession preserving equivalence that is the identity plus that single equivalence.
However, I can’t build and/or find the equivalence relation such that forall a, b in N a+b=b+a. I can neither build it or find it ( the difference between these options is moot from within the theory), however much I suspect that I could build it, or however much I pile up a finite number of examples that it works.
However, this exact succession-preserving equivalence relation is readily proved, i.e. build or found, in the full Peano arithmetic. (I can still build all the ones I already had.)
In that sense the Peano arithmetic has more objects (in this model, succession-preserving equivalence relations) than the weaker one.
I would very much like to get rid of uncountable anything from a fully functioning theory of mathematics. Ideally, it would encompass both algebra AND analysis. Maybe I’m losing it, but I’m starting to think it might be doable to construct such a theory.
Some cherished ideas in math would have to be thrown out or reinterpreted (e.g Cantor’s diagonalization proof), but you would get rid of a bunch of paradoxes too.
There can be two square roots of -1, you would just need two separate proofs (probably isomorphic, depending on the exact proof system you are using) to produce each one. In essence, this is what we do in practice anyway: we prove the value of the positive square root, and by independent principle from there, that the negative root also exists. (We can of course do it the other way round, hence why it is isomorphic.)
My view on uncountables is admittedly unorthodox (though by no means unique to me), however the proof theoretic stuff I’m describing here is pretty vanilla.
I’m not sure what you mean here, but computer scientists seem to like to work with typed lambda calculi (whose categorical semantics are cartesian closed categories). Suppose a typed lambda calculus has a type N for natural numbers (a countable type), and let A be any type. Then if A^N is countable (if there exists an isomorphism N –> A^N), one can use this isomorphism to cook up an explicit fixed-point combinator Y: A^A –> A. In particular, taking A = N, we conclude that successor s: N –> N has a fixed point Y(s) in N.
How would you respond to that?
Being ultimately of the computer science persuasion myself, this is the direction I’m going in.
Ultimately, I want to move towards a “computable math” where all objects are either finite or “well-behaved” infinities. In the end, I think this is what constructivism is all about.
My interest in the foundations of mathematics is more of an avocation, but my feeling (or hope) is that these computational approaches can be extended to all of traditional math, perhaps requiring some concessions in the translation.
I think one of the challenges in this regard is exactly restricting the exponentials to be countable in an intuitive and manageable way in categories (or versions thereof) that are more natural or palatable to non-computational mathematicians.
I think part of the problem may be that people have way too much fun with the uncountables.
One thing I wonder about: apart from getting rid of “paradoxes”, is there ultimately any benefit in throwing away most of math “as we know it” and translating it into some constructive framework?
The hope is that most math as we know it doesn’t need to be thrown away, just tightened up a bit, or put on firmer ground. (Some proofs may in fact become easier, since you wouldn’t have to worry about uncountables.)
The advantage is having higher confidence that our proofs are actually consistent and that the system as a whole makes sense. Mathematicians worrying about uncountables and the problems they cause might be like biologists worrying about the impact on the ecology of purple unicorns. (And possibly worse: proving wacky theorems about the common cat based on the characteristics of purple unicorns.)
To put this in perspective, consider the fact, hinted in one my previous comments in this thread, that when it comes to proofs everybody is a constructivist.
If I told you that I had an existence proof that proved there was a proof for propostition P (whatever that may be) but my proof didn’t tell me how to actually build the proof, would you accept P as proven?
If you study any proof theory to any depth, you start to realize that building a proof and building a mathematical object with desired properties is pretty much exactly the same thing.
So why believe in uncountable sets when even an infinite sequence of finite induction steps can’t build/define one clearly enough to actually reason about? (Just because Cantor convinced you that if he had an infinity of infinities he could produce a purple unicorn?)
Marc, maybe I wasn’t being clear; I was trying to point out a potential problem. Let me try again.
Suppose given a model of typed lambda calculus with a natural numbers object N, such that A^N is countable (isomorphic to N) for some type A. Let G: A^N –> N be an isomorphism, with inverse H. Then there is a fixed point combinator Y: A^A –> A, defined by the formulas
t = G(lambda n. f(H(n)(n))), Y(f) = H(t)(t)
so that every f: A –> A has a fixed point Y(f). The proof is manifestly constructive.
But now, suppose we now apply it to A = a type of truth values, and the map A –> A is taken to be negation. Then we get a term t of type A (i.e., a truth value) equal to its own negation. Then t = t /\ t = t /\ not(t) = 0. So t = 0 is equal to its own negation, 1. Therefore 0 = 1.
Something’s gotta give… the most common response to this constructive version of Cantor’s diagonal argument would be to say that internally (i.e., within the model), A^N cannot be countable for A = truth-value-type, or for any type A which has a fixed-point-free endomorphism (like s: N –> N).
I was wondering how you would respond to that.
There are “imaginary logics” with truth values I and J such that each is its own negation, with a time-delay, or phase shift, or other modification. Kauffman (best known for his Knot Theory) is one author who’s treated these.
For example,
I = …0101010101010…
J = …1010101010101…
oscillating between 1 = True and 0 = False. I have a draft paper which generalizes this from 2 “imaginary” truth values to integer N.
The logic with I and J gives a circuit for “square root of NOT” different from the Quantum Computing design, and my generalization gives Nth root of NOT.
Dan Piponi has expressed some interest in building a programming language around such “imaginary logics” (a term invented by Vasiliev at the start of the 20th century, and more recently reintropduced by those giving modern formalizations to the wilder aspects of his work).
Sorry, Jonathan, I must be missing something. Your J looks like not(I) to me. Are you saying I and J are equal?
What logic with I and J? Is there a conjunction? What axioms are satisfied?
Todd Trimble:
Busy as I am, about to start classes tomorrow at grad school for the first time in 34 years, if you email me at jvospost2(AT)yahoo(DOT)com, I can email you (or others who ask) the draft paper in question, which is extensive in summarizing and citing the literature on Imaginary Logic, most of which do use the Boolean Logic axioms but with the extended set of truth values.
The question behind your question is, for:
I = …0101010101010…
J = …1010101010101…
is NOT(I) = I? is NOT(J) = J?
It depends on how one handles the phase shift.
Still not getting it. You said “truth values I and J such that each is its own negation”, and so I was following your lead by taking
not(I) = I, not(J) = J
as assumptions. But then, since J looks like not(I) to me (I’m just guessing, of course) I seem to conclude I = J.
Feel free to explain whenever you have the time.
You are assuming a subobject classifier that is classical; by controlling the endomorphisms on that object you can and their compositions you can alter its behaviour. The ordinary intuitionistic subobject classifier or a variant should work here. This would correspond to non-termination of the proof that constructs these things on some inputs, i.e. some of these mappings must be partial. This is pretty much standard comp sci/ proof theory, no? Or is there some tricky ramification you are trying to point out?
A different solution (not incompatible with the previous one) is that we think of N as a kind of self-limit at infinity. (If we are going to get Cauchy sequences in here for our analysis friends, we are going to have such a notion of limits anyway) In that sense, the successor function (suitably defined in the order theoretic way) DOES have a least fix point (at infinity) and that limit is what makes N an acceptable constructive entity in the first place.
None of this is controversial with respect to compuational math (not that I can see anyway, unless you think I’ve missed something); the only innovation I’m proposing is finding way to do anything you might want to in analysis (or any other branch of math) in a convenient way using only the computable suborders and permutations of N, with the addition of such limits.
(You see why I was interested in the constructive topos thread…)
No, I’m not. I was careful to write my argument so that it is intuitionistically valid.
No. For those who are having trouble following this, let me recall the notion of Heyting algebra (the standard intuitionistic variant of Boolean algebra). It’s a lattice H (so, it has all finite meets and finite joins; 1 is the empty meet and 0 is the empty join), with an implication operator b => c such that (for all a, b, c)
a |- (b => c) iff a /\ b |- c
where |- denotes the partial order (entailment) of H. This property says roughly that b => c is the weakest assumption whose conjunction with b entails c.
The negation operator on a Heyting algebra is defined to be
not(b) := (b => 0).
I claim not(0) = 1. This is equivalent to (for all a) a |- 0 => 0, which by the property above is equivalent to a /\ 0 |- 0. Which is true, since a /\ b |- b for any b.
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.
Finally, let me point out that while a data type N* (N plus infinity) could be an interesting surrogate for the ordinary N, there is a standard meaning of “natural number type”, such that a fixed point for successor would imply that every endomorphism A –> A for every type A has a fixed point. That’s not necessarily devastating (examples of such categories crop up in domain theory, certainly), but there are then attendant severe restrictions on what data types are admissible (e.g., truth value types), and maybe severe restrictions on how much analysis you can actually do.
I think I now understand the claim that a topos is not the setting for a constructive set theory.
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?
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).
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!).
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.
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.
Todd Trimble:
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.
Well, thanks, Jonathan; you needn’t have gone to such lengths.
I am familiar with a number of items in your list (e.g., Peirce, Lawvere, Sloane, Street); AFAICT the ones which are *directly* relevant to your post are some of the ones by Kauffman, particularly “Time, Imaginary Value, Paradox, Sign and Space”, which you handily gave a link for — I’ll take a closer look.
Interesting that Kauffman and Varela were once co-authors!
Jonathan, I did take a preliminary look at the paper by Kauffman, and also another long draft of a paper of Kauffman which looked interesting, a long meditation on Spencer-Brown’s Laws of Form, a book he mentions in his Knot Theory. Funny, I never noticed before but was struck now by the similarity between Spencer-Brown’s Calculus of Indications and that fragment of Peirce’s Existential Graphs known as system Alpha (which is a graphical calculus for propositional logic, i.e., for free Boolean algebras).
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?
Well, I can ask him tomorrow morning…
From Kauffman:
“The answer is: I have never met the Reverend Flagg. James Flagg live in LA and is involved in airport design. He and I have been discussing mathematics for almost 30 years.”
So that’s one sane mathematician. Now to verify the others…
Todd Trimble:
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?