
How can you resist a book with a title like "Inconsistent Mathematics"?
1) Chris Mortensen, Inconsistent Mathematics, Kluwer, Dordrecht, 1995.
Ever since Goedel showed that all sufficiently strong systems formulated using the predicate calculus must either be inconsistent or incomplete, most people have chosen what they perceive as the lesser of two evils: accepting incompleteness to save mathematics from inconsistency. But what about the other option?
This book begins with the startling sentence: "The following idea has recently been gaining support: that the world is or might be inconsistent." As we reel in shock, Mortensen continues:
Let us consider set theory first. The most natural set theory to adopt is undoubtedly one which has unrestricted set abstraction (also known as naive comprehension). This is the natural principle which declares that to every property there is a unique set of things having the property. But, as Russell showed, this leads rapidly to the contradiction that the the Russell set [the set of all sets that do not contain themselves as a member] both is and is not a member of itself. The overwhelming majority of logicians took the view that this contradiction required a weakening of unrestricted abstraction in order to ensure a consistent set theory, which was in turn seen as necessary to provide a consistent foundation for mathematics. But all ensuing attempts at weakening set abstraction proved to be in various ways ad hoc. Da Costa and Routley both suggested instead that the Russell set might be dealt with more naturally in an inconsistent but nontrivial set theory (where triviality means that every sentence is provable).An inconsistent but nontrivial logical system is called paraconsistent. But it's not so easy to create such systems. To keep an inconsistency from infecting the whole system and making it trivial, we need to drop the rule of classical logic which says that "A and not(A) implies B" for all propositions A and B. Unfortunately, this rule is built into the propositional calculus from the very start!
So, we need to revise the propositional calculus.
One way to do it is to abandon "material implication"  the form of implication where you can calculate the truth value of "P implies Q" from those of P and Q using the following truth table:
P  Q  P implies Q  T  T  T T  F  F F  T  T F  F  TWith material implication, a false statement implies every statement, so any inconsistency is fatal. But in real life, if we discover we have one inconsistent belief, we don't conclude we can fly and go jump off a building! Material implication is really just our best attempt to define implication using truth tables with 2 truth values: true and false. So it's not surprising that logicians have investigated other forms of implication.
One obvious approach is to use more truth values, like "true", "false", and "don't know". There's a long history of work on such multivalued logics.
Another approach, initiated by Anderson and Belnap, is called "relevance logic". In relevance logic, "P implies Q" can only be true if there is a conceptual connection between P and Q. So if B has nothing to do with A, we don't get "A and not(A) implies B".
This book describes a logical system called "RQ"  relevance logic with quantifiers. It also describes a system called "R#", which is a version of the Peano axioms of arithmetic based on RQ instead of the usual predicate calculus. Following the work of Robert Meyer, it proves that R# is nontrivial in the sense described above. Moreover, this proof can be carried out R# itself! However, you can carry out the proof of Goedel's 2nd incompleteness theorem in R#, so R# cannot prove itself consistent.
To paraphrase Mortensen: "But this is not really a puzzle. The explanation is that relevant and other paraconsistent logics turn on making a distinction between inconsistency and triviality, the former being weaker than the latter; whereas classical logical cannot make this distinction. For what the present author's intuitions are worth, these do seem to be different concpets. Thus for R#, consistency cannot be proved by finitistic means by Goedel's second theorem, whereas nontriviality can be shown. Since Peano arithmetic collapses this distinction, both kinds of consistency are infected by the same unprovability."
Mortensen also mentions another approach to get rid of "A and not(A) implies B" without getting rid of material implication. This is to get rid of the rule that "A and not(A)" is false! He calls this "Brazilian logic". Presumably this is not because your average Brazilian thinks this way, but because the inventor of this approach, Da Costa, is Brazilian.
Brazilian logic sounds very bizarre at first, but in fact it's just the dual of intuitionistic logic, where you drop the rule that "A or not(A)" is true. Intuitionistic logic is nicely modeled by open sets in a topological space: "and" is intersection, "or" is union, and "not" is the interior of the complement. Similarly, Brazilian logic is modeled by closed sets. In intuitionistic logic we allow a slight gap between A and not(A); in Brazilian logic we allow a slight overlap.
In short, this book is full of fascinating stuff. Lots of passages are downright amusing at first, like this:
[...] there have been calls recently for inconsistent calculus, appealing to the history of calculus in which inconsistent claims abound, especially about infinitesimals (Newton, Leibniz, Bernoulli, l'Hospital, even Cauchy). However, inconsistent calculus has resisted development.But you always have to remember that the author is interested in theories which, though inconsistent, are still paraconsistent. And I think he really makes a good case for his claim that inconsistent mathematics is worth studying  even if our ultimate goal is to avoid inconsistency!
Okay, now let me switch gears drastically and say a bit about "exotic spheres"  smooth manifolds that are homeomorphic but not diffeomorphic to the nsphere with its usual smooth structure. People on sci.physics.research have been talking about this stuff lately, so it seems like a good time for a miniessay on the subject. Also, my colleague Fred Wilhelm works on the geometry of exotic spheres, and he just gave a talk on it here at U. C. Riverside, so I should pass along some of his wisdom while I still remember it.
First, recall the "Hopf bundle". It's easy to describe starting with the complex numbers. The unit vectors in C^2 form the sphere S^{3}. The unit complex numbers form a group under multiplication. As a manifold this is just the circle S^{1}, but as a group it's better known as U(1). You can multiply a unit vector by a unit complex number and get a new unit vector, so S^{1} acts on S^{3}. The quotient space is the complex projective space CP^{1}, which is just the sphere S^{2}. So what we've got here is fiber bundle:
S^{1}→ S^{3} → S^{2} = CP^{1}
with fiber S^{1}, total space S^{3} and base space S^{2}. This is the Hopf bundle. It's famous because the map from the total space to the base was the first example of a topologically nontrivial map from a sphere to a sphere of lower dimension. In the lingo of homotopy theory, we say it's the generator of the group π_{3}(S^{2}).
Now in "week106" I talked about how we can mimic this construction by replacing the complex numbers with any other division algebra. If we use the real numbers we get a fiber bundle
S^{0} → S^{1} → RP^{1} = S^{1}
where S^{0} is the group of unit real numbers, better known as Z/2. This bundle looks like the edge of a Moebius strip. If we use the quaternions we get a more interesting fiber bundle:
S^{3} → S^{7} → HP^{1} = S^{4}
where S^{3} is the group of unit quaternions, better known as SU(2). We can even do something like this with the octonions, and we get a fiber bundle
S^{7}→ S^{15} → OP^{1} = S^{8}
but now S^{7}, the unit octonions, doesn't form a group  because the octonions aren't associative.
Anyway, it's the quaternionic version of the Hopf bundle that serves as the inspiration for Milnor's construction of exotic 7spheres. These exotic 7spheres are actually total spaces of other bundles with fiber S^{3} and base space S^{4}. The easiest way to get your hands on these bundles is to take S^{4}, chop it in half along the equator, put a trivial S^{3}bundle over each hemisphere, and then glue these together. To glue these bundles together we need a way to attach the fibers over each point x of the equator. In other words, for each point x in the equator of S^{4} we need a map
f_{x}: S^{3} → S^{3}
which should vary smoothly with x. But the equator of S^{4} is just S^{3}, and S^{3} is a group  the unit quaternions  so we can take
f_{x}(y) = x^{n} y x^{m}
for any pair of integers (n,m).
This gives us a bunch of S^{3}bundles over S^{4}. The total space X(n,m) of any one of these bundles is obviously a smooth 7dimensional manifold. But when is it homeomorphic to the 7sphere? And when is it diffeomorphic to the 7sphere with its usual smooth structure?
Well, first we use some Morse theory. You can learn a lot about the topology of a smooth manifold if you have a "Morse function" on the manifold: a smooth realvalued function all of whose critical points are nondegenerate. If you don't believe me, read this book:
2) John Milnor, Morse Theory, Princeton U. Press, Princeton, 1960.
When n + m = 1 there's a Morse function on X(n,m) with only two critical points  a maximum and a minimum. This implies that X(n,m) is homeomorphic to a sphere!
Once we know that X(n,m) is homeomorphic to S^{7}, we have to decide when it's diffeomorphic to S^{7} with its usual smooth structure. This is the hard part. Notice that X(n,m) is the unit sphere bundle of a vector bundle over S^{4} whose fiber is the quaternions. We can understand a bunch about X(n,m) using the characteristic classes of this vector bundle. In particular, we can compute the Euler number and the Pontrjagin number of this vector bundle. Using the Euler number we can show that X(n,m) is homeomorphic to a sphere only if n + m = 1  you can't really do this using Morse theory. But more importantly, using the Pontrjagin number, we can show that in this case X(n,m) is diffeomorphic to S^{7} with its usual smooth structure if and only if (n  m)^{2} = 1 mod 7. Otherwise it's "exotic".
For the details of the above argument you can try the following book:
3) B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry  Methods and Applications, Part III: Introduction to Homology Theory, SpringerVerlag Graduate Texts, number 125, Springer, New York, 1990.
or the original paper:
4) John Milnor, On manifolds homeomorphic to the 7sphere, Ann. Math 64 (1956), 399405.
Now, with quite a bit more work, you can show that smooth structures on the nsphere form an group under connected sum  the operation of chopping out a small hole in two spheres and gluing them together  and you can show that this group is Z/28 for n = 7. This means that if we consider two smooth structures on the 7sphere the same when they're related by an orientationpreserving diffeomorphism, we get exactly 28 kinds. Unfortunately we don't get all of them by the above explicit construction. For more details, see:
5) M. Kervaire and J. Milnor, Groups of homotopy spheres I, Ann. Math. 77 (1963), 504537.
By the way, part II of the above paper doesn't exist! Instead, you should read this:
6) J. Levine, Lectures on groups of homotopy spheres, in Algebraic and Geometric Topology, Springer Lecture Notes in Mathematics number 1126, Springer, Berlin, 1985, pp. 6295.
Anyway, if you're wondering why I'm talking so much about exotic 7spheres, instead of lowerdimensional examples that are easier to visualize, check out this table of groups of smooth structures on the nsphere:
n group of smooth structures on the nsphere 0 1 1 1 2 1 3 1 4 ? 5 1 6 1 7 Z/28 8 Z/2 9 Z/2 x Z/2 x Z/2 10 Z/6 11 Z/992 12 1 13 Z/3 14 Z/2 15 Z/8128 x Z/2 16 Z/2 17 Z/2 x Z/2 x Z/2 x Z/2 18 Z/8 x Z/2Dimension 7 is the simplest interesting case  except perhaps for dimension 4, where the answer is unknown! The "smooth Poincare conjecture" says that there's only one smooth structure on the 4sphere, but this remains a conjecture....
As you can see, there are lots of exotic 11spheres. In fact, this is relevant to string theory! You can get an nsphere with any possible smooth structure by taking two ndimensional balls and gluing them together along their boundary using some orientationpreserving diffeomorphism
f: S^{n1} → S^{n1}.
Orientationpreserving diffeomorphisms like this form a group called Diff_{+}(S^{n1}). Using the above trick, it turns out that the group of smooth structures on the nsphere is isomorphic to the group of connected components of Diff_{+}(S^{n1}). So the existence of exotic 11spheres means that there are lots of "exotic diffeomorphisms" of the 10sphere!
Now, string theory lives in 10 dimensions, and one wants certain quantities to be invariant under orientationpreserving diffeomorphisms of spacetime  otherwise you say the theory has "gravitational anomalies". First you have to check this for "small diffeomorphisms" of spacetime, that is, those connected to the identity map by a continuous path. But then you have to check it for "large diffeomorphisms"  those living in different connected components of the diffeomorphism group. When spacetime is a 10sphere, this means you need to check diffeomorphism invariance for all 991 components of Diff_{+}(S^{n1}) besides the component containing the identity. These components correspond to exotic 11spheres!
Witten did this in the following paper:
7) Edward Witten, Global gravitational anomalies, Commun. Math. Phys. 100 (1985), 197229.
This may be the first paper about exotic spheres in physics.
There are other interesting things to do with an exotic sphere. One is to put a metric on it and look at its curvature. The sphere with its usual "round" metric is very symmetrical and has positive curvature everywhere. There are various meanings of "positive curvature", but the round sphere has positive curvature in all possible ways! One kind of curvature is "sectional curvature". In general, it's hard to find compact manifolds other than the sphere with its usual smooth structure that have metrics with everywhere positive sectional curvature. Gromoll and Meyer found an exotic 7sphere with a metric having nonnegative sectional curvature:
8) Detlef Gromoll and Wolfgang Meyer, An exotic sphere with nonnegative sectional curvature, Ann. Math. 100 (1974), 401406.
The construction isn't terribly hard so let me describe it. First, start with the group Sp(2), consisting of 2x2 unitary quaternionic matrices (see "week64"). As always with compact Lie groups, this has a metric that's invariant under right and left translations, and this metric is unique up to a constant scale factor. The group of unit quaternions acts as metricpreserving maps (aka "isometries") of Sp(2) in the following way: let the quaternion q map
(a b) (c d)to
(qaq^{1} qb) (qcq^{1} qd)The quotient space is an exotic 7sphere, and it inherits a metric with nonnegative sectional curvature.
Now, since compact manifolds with positive sectional curvature are tough to find, you might wonder if this exotic 7sphere can be given a metric with positive sectional curvature. And the answer is: almost! It can be given a metric having positive sectional curvature except on a set of measure zero. This was recently proved by Wilhelm:
9) Frederick Wilhelm, An exotic sphere with positive curvature almost everywhere, preprint, May 12 1999.
It's also an interesting theorem, due to Hitchin, that for any n > 0 there exist exotic spheres of dimensions 8n+1 and 8n+2 having no metric of positive scalar curvature:
8) Nigel Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 155.
So some exotic spheres are not so as "round" as you might think! In fact, 3 of the exotic spheres in 10 dimensions cannot be given a metric such that the connected component of the isometry group is bigger than U(1) x U(1), so these are quite "bumpy". This follows from results of Reinhard Schultz, who happens to be the department chair here:
9) Reinhard Schultz, Circle actions on homotopy spheres bounding plumbing manifolds, Proc. A.M.S. 36 (1972), 297300.
There's a lot more to say about exotic spheres, but let me just briefly mention two things. First, there are cool connections between exotic spheres and higherdimensional knot theory. If you want a small taste of this stuff, try:
10) Louis Kauffman, Knots and Physics, World Scientific, Singapore, 1991.
Look in the index under "exotic spheres".
Second, people have computed the effect of exotic 7spheres on quantum gravity path integrals in 7 dimensions:
11) Kristin Schleich and Donald Witt, Exotic spaces in quantum gravity, Class. Quant. Grav. 16 (1999) 24472469, preprint available as grqc/9903086.
I'm not sure exotic spheres are really relevant to physics, but it would be cool, so I'm glad some people are trying to establish connections.
Okay, that's enough for exotic spheres, at least for now! I've got a few more things here that I just want to mention....
I've been learning a bit about CalabiYau manifolds and mirror symmetry in string theory lately. The basic idea is that string theory on different spacetime manifolds can be physically equivalent. I don't know enough to want to try to explain this stuff yet, but here are some place to look in case you're interested:
12) Claire Voisin, Mirror Symmetry, American Mathematical Society, 1999.
13) David A. Cox and Sheldon Katz, Mirror Symmetry and Algebraic Geometry, American Mathematical Society, Providence, Rhode Island, 1999.
14) ShingTung Yau, editor, Mirror Symmetry I, American Mathematical Society, 1998.
Brian Green and ShingTung Yau, editors, Mirror Symmetry II, American Mathematical Society, 1997.
Duong H. Phong, Luc Vinet and ShingTung Yau, editors, Mirror Symmetry III, American Mathematical Society, 1999.
So far I'm mainly trying to learn really basic stuff, and for this, the following lectures are proving handy:
16) P. Candelas, Lectures on complex manifolds, in Superstrings '87, eds. L. AlvarezGaume et al, World Scientific, Singapore, 1988, pp. 188.
On a different note, the American Mathematical Society has come out with some goodlooking books on surgery theory  the process of making new manifolds from old by cutting and pasting. I've got these on my reading list, so if anyone wants to buy me a Christmas present, here's what you should get:
17) Robert E. Gompf and Andras I Stipsicz, 4Manifolds and Kirby Calculus, Amderican Mathematical Society, 1999.
18) C. T. C. Wall and A. A. Ranicki, Surgery on Compact Manifolds, 2nd edition, American Mathematical Society, 1999.
Finally, there's some cool stuff going on with operads that I haven't been able to keep up with. Let me quote the abstracts:
19) Alexander A. Voronov, Homotopy Gerstenhaber algebras, preprint available as math.QA/9908040.
The purpose of this paper is to complete GetzlerJones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B_inftyoperad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a B_inftyalgebra, another, called a homotopy Galgebra, is a particular case of a B_inftyalgebra, the others, a G_inftyalgebra, an E^1baralgebra, and a weak G_inftyalgebra, arise from the geometry of configuration spaces. Corrections to the paper math.QA/9602009 of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made.
20) Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 3572, preprint available as math.QA/9904055.
It became clear during last 56 years that the algebraic world of associative algebras (abelian categories, triangulated categories, etc) has many deep connections with the geometric world of twodimensional surfaces. One of the manifestations of this is Deligne's conjecture (1993) which says that on the cohomological Hochschild complex of any associative algebra naturally acts the operad of singular chains in the little discs operad. Recently D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higherdimensional algebras and of motives to quantum field theories.
21) James E. McClure and Jeffrey H. Smith, A solution of Deligne's conjecture, preprint available as math.QA/9910126
Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action of an operad equivalent to the little 2cubes.
My original table of groups of smooth structures on spheres had some mistakes in it which were corrected by Linus Kramer, Marco Mackaay, Tony Smith and Frederick Wilhelm. In fact, the table in the book by Dubrovin, Fomenko and Novikov differs from the table in Kervaire and Milnor's paper! The table above comes from Kervaire and Milnor, taking advantage of some subsequent work in dimension 3 and also some work of Brumfield which nailed down the groups in dimensions 9 and 17  see below for more information.
The paper by Kervaire and Milnor has a cool formula for the order of the group of smooth structures on the (4n1)sphere for n > 1. It's:
2^(2n4) (2^(2n1)  1) P(4n1) B(n) a(n) / n
where:
P(k) is the order of the kth stable homotopy group of spheres
B(k) is the kth Bernoulli number, in the sequence 1/6, 1/30, 1/42, 1/30, 5/66, 691/2730, 7/6, ...
a(k) is 1 or 2 according to whether k is even or odd.
Here are some remarks by Linus Kramer on exotic spheres in dimensions 9 and 17, which he posted to sci.math.research in response to a question of mine. Kervaire and Milnor said the group of exotic spheres in dimension 9 was (Z/2)^{3} or Z/2 x Z/4, and the group in dimension 17 was (Z/2)^{4} or (Z/2)^{2} x Z/4. Linus writes:
The list by Kervaire and Milnor seems to be correct; in dimension 9, the group is (Z/2)^3, and in dimension 17 it's (Z/2)^4. This follows from the results of Brumfield [Mich. Math. J 17] stated on the first page of his paper, plus the list of the first stable homotopy groups of spheres, and the properties of Adams' Jhomomorphism J:π_n(SO) → π^s_n. There is an exact sequence 0 → bP_k → \Gamma_{k1} → \pi^s_{k1}/im(J) → 0 provided that k+3 is not a power of 2 (\Gamma_{k1} is the group we are looking for). Now for k1=8,9,10,17, we have k+3=12,13,14,21, and these are not powers of 2. Now bP_k=0,Z/2,0,Z/\theta_{16} for k=9,10,11,16. So for k=9,11, the group \Gamma_{k1} is the same as \pi^s_{k1}/im(J), and for k=10,16 use Brumfield's result that then the group is \Gamma_{k1} = Z/2+(\pi^s_{k1}/im(J)). Hope I didn't make a mistake while chasing through all these exact sequences... Of course, the result relies also on some table, namely the first stable homotopy groups of spheres. Linus Kramer
