**Stephen P. King** (*stephenk1@home.com*)

*Wed, 27 Oct 1999 14:54:08 -0400*

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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 | T

With 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 multi-valued

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 n-sphere 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 mini-essay 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 pi_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 7-spheres. These

exotic 7-spheres 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 7-dimensional manifold.

But when is it homeomorphic to the 7-sphere? And when is it *diffeomorphic*

to the 7-sphere 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 real-valued 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,

Springer-Verlag Graduate Texts, number 125, Springer, New York, 1990.

or the original paper:

4) John Milnor, On manifolds homeomorphic to the 7-sphere, Ann.

Math 64 (1956), 399-405.

Now, with quite a bit more work, you can show that smooth structures on

the n-sphere 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 7-sphere the same when they're related by

an *orientation-preserving* 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), 504-537.

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. 62-95.

Anyway, if you're wondering why I'm talking so much about exotic 7-spheres,

instead of lower-dimensional examples that are easier to visualize, check

out this table of groups of smooth structures on the n-sphere:

n group of smooth structures on the n-sphere

0 1

1 1

2 1

3 1

4 1

5 1

6 1

7 Z/28

8 Z/2

9 Z/2 x Z/2

10 Z/2 x Z/2 x Z/2

11 Z/992

Dimension 7 is the simplest interesting case!

As you can see, there are lots of exotic 11-spheres. In fact, this is

relevant to string theory! You can get an n-sphere with any possible

smooth structure by taking two n-dimensional balls and gluing them together

along their boundary using some orientation-preserving diffeomorphism

f: S^{n-1} -> S^{n-1}.

Orientation-preserving diffeomorphisms like this form a group called

Diff_+(S^{n-1}). Using the above trick, it turns out that the group of

smooth structures on the n-sphere is isomorphic to the group of *connected

components* of Diff_+(S^{n-1}). So the existence of exotic 11-spheres

means that there are lots of "exotic diffeomorphisms" of the 10-sphere!

Now, string theory lives in 10 dimensions, and one wants certain quantities

to be invariant under orientation-preserving 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 10-sphere, this

means you need to check diffeomorphism invariance for all 991 components of

Diff_+(S^{n-1}) besides the component containing the identity. These

components correspond to exotic 11-spheres!

Witten did this in the following paper:

6) Edward Witten, Global gravitational anomalies, Commun. Math. Phys.

117 (1986), 197-229.

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 7-sphere with a metric having *nonnegative*

sectional curvature:

6) Detlef Gromoll and Wolfgang Meyer, An exotic sphere with nonnegative

sectional curvature, Ann. Math. 100 (1974), 401-406.

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 metric-preserving 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 7-sphere, 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 7-sphere can be given

a metric with *positive* sectional curvature. And the answer is: yes!

This was recently proved by Wilhelm:

7) 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), 1-55.

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), 297-300.

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 higher-dimensional 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 7-spheres on

quantum gravity path integrals in 7 dimensions:

11) Kristin Schleich and Donald Witt, Exotic spaces in quantum

gravity, Class. Quant. Grav. 16 (1999) 2447-2469, preprint available

as gr-qc/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 Calabi-Yau 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) Shing-Tung Yau, editor, Mirror Symmetry I, American Mathematical

Society, 1998.

Brian Green and Shing-Tung Yau, editors, Mirror Symmetry II, American

Mathematical Society, 1997.

Duong H. Phong, Luc Vinet and Shing-Tung 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. Alvarez-Gaume et al, World Scientific, Singapore, 1988, pp. 1-88.

On a different note, the American Mathematical Society has come out

with some good-looking 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, 4-Manifolds 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 Getzler-Jones' 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_infty-operad, 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_infty-algebra,

another, called a homotopy G-algebra, is a particular case of a

B_infty-algebra, the others, a G_infty-algebra, an E^1-bar-algebra, and

a weak G_infty-algebra, 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), 35-72, preprint available as math.QA/9904055.

It became clear during last 5-6 years that the algebraic world of

associative algebras (abelian categories, triangulated categories, etc)

has many deep connections with the geometric world of two-dimensional

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 higher-dimensional algebras and of

motives to quantum field theories.

-----------------------------------------------------------------------

Previous issues of "This Week's Finds" and other expository articles on

mathematics and physics, as well as some of my research papers, can be

obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twf.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html

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