My adviser, and many other folks besides, seem to have major beef with proof by contradiction.  It works, but it doesn’t advance the field.  Digging into a proof by contradiction, you’re not going to find a useful object to study.

But I find proofs by contradiction fascinating, and they feel quite natural to me somehow.  So I want to write a book of them.  More specifically, I want to see how many standard and/or famous proofs by contradiction can be interestingly tweaked into the ur-contradiction 0=1.   I’ll call it Conjectures on the One-Element Field.

Most blow-up arguments (like Sesum’s that I just posted about) fall into this category, as do a lot of index-theory arguments (the Hairy Ball Theorem needs its own chapter, I think).  The trick is to make the 0=1 part flow from the rest the proof, and not to be too obvious and artificial about it.

To get the proof to work in this case, I note that an inequality like is how one establishes Glickenstein’s Lemma.  If , this inequality is immediate since .

However, we can still get such an inequality even if isn’t bounded.  First note that, for the rescaled flows , we have a uniform Ricci bound .  So for each we can choose close enough to that

Here ‘close enough’ is independent of .  Then, using the definition of , we see that

So if we want to prove, for any and , , we just have to write , which can clearly be done.  In fact we just have to find one such .  So we get a Glickenstein’s Lemma for a flow with merely bounded.

The rest of the proof works just fine, because in the limit we still get , hence by the evolution equation

we see that .

Recall that we’re dealing with a family of 2-convex hypersurfaces evolving via mean curvature flow.  We’ll assume uniform 2-convexity, i.e. for some .

I won’t go into the analytic work, but we need to use it.  There are two analytic results we need:

Cylindrical Estimate. For any , there exists also depending on and the diameter of the initial hypersurface, so that along the flow

I call this a cylindrical estimate because the quantity vanishes on a cylinder, and when is small it measures the sum of the squares of the differences of the higher eigenvalues.

Harnack Estimate. For any , there exist also depending on , the initial diameter, so that along the flow

One first establishes these estimates for the smooth flow, and it’s rather easy to see that the surgery procedure doesn’t affect the constants, since surgery involves replacing approximate cylinders with approximate spheres.

Recall that the surgery procedure relies upon finding Hamilton necks–that is, regions of  the hypersurface that are approximate round cylinders.  In fact we want to do better than finding a region which looks like a cylinder at one time; we want to find regions that shrink like cylinders under the flow.

Definition. We call the spacetime region an shrinking neck if

• The final time-slice is a Hamilton neck  -close to a round neck in .  Call the approximate radius of this neck .
• is close to , where is the radius of a round cylinder at time , which shrinks to radius at time .
• , and each neck has length .

All of the singularity analysis resides in the following theorem:

Neck Detection Lemma. Let be constants as above.  There exist constants depending on , and the initial diameter, so that if any satisfies:

1. and
2. The parabolic neighbourhood doesn’t contain any points affect by a previous surgery.

then is an -shrinking neck.

To prove this theorem, we are going to argue by contradiction, using a blow-up argument.

Proof of the Neck Detection Lemma.  If the theorem is not true, we may take a sequence of flows, each of which contains a point so that and each parabolic neighbourhood is unaffected by surgery, and so that none of them has as a shrinking neck.

Rescale and translate each of the flows: to get so that .  The parabolic neighbourhoods all  rescale to the same .

Tracing the Harnack estimate,  we get a Harnack estimate for , so that the are bounded uniformly on .  Then the cylindrical estimate says that are uniformly bounded in .

Then the Harnack estimate says that in fact all derivatives of $latex\tilde{A}_j$ are uniformly bounded in .  So according to the Arzela-Ascoli theorem, there is a limit flow , with convergence in where is the control we have on the derivatives during surgery.

Consider the properties of . We have .

Now ,  and satisfies a parabolic equation, so achieving its minimum at an interior point means in fact .  Then .  So in fact has a zero eigenvalue and all other eigenvalues equal.

The evolution equation for the quantity (which we have just proved is identically zero) has a term involving , so we see that , i.e. is covariant constant.  A rigidity theorem of Lawson says that this characterises cylinders.

Thus is a shrinking cylinder on .  So the sequence is approaching a shrinking cylinder, hence for large enough , is a shrinking neck.  But is just a rescaling of .  So is a shrinking neck, in contradiction to our assumption.

Characterisation of Ricci Flow Singularities.  Suppose is a Ricci flow, defined up to time .  Assume is maximal.  Then we have , where is the Riemann tensor.

That is to say, what goes wrong at a Ricci flow singularity is the curvature tensor.  We could restate the theorem as its contrapositive:

Characterisation of When the Flow Can Be Extended. Suppose that up to time , we have a bound .  Then the Ricci flow can be extended past .

But of course the curvature tensor is a big nasty gadget, which is none too easy to understand.  We want something easier to check than “Are my curvatures all bounded?”.  Šešum proved that, in fact, one need only look at the trace of the Riemann tensor, which is the Ricci tensor:

Better Characterisation of Ricci Flow Singularities. As above, let be the first singular time of a Ricci flow.  Then .

This is somewhat remarkable, because the Ricci tensor interacts weirdly with Ricci flow.  For example, the condition is not preserved under the flow.  Yet somehow the Ricci tensor carries the information about when singularities occur.  Proof under the fold.

We start with a ball-collapse estimate.  Ultimately, the estimate derives from Perel’man’s density ideas.

Glickenstein’s Lemma.  Let .  Then we have , where is a bound for the Ricci tensor.

That is to say, balls are collapsing at worst like .

Now suppose that we have a flow with , but which has a singularity at .  We will obtain a contradiction by a blow-up argument.

Since , is blowing up, i.e. there is a sequence with , .  Set .  Consider the rescaled metrics

each of which is also a solution to the Ricci flow.  The pointed manifolds converge to a limit for each .  By the work of Hamilton, is an ancient solution to the Ricci flow.

We want to come up with more properties of .  Since , we have as well.  The rescaled scalar curvature is , and since , we see that

Under Ricci flow, the scalar curvature evolves as , so the vanishing of implies the vanishing of .  Thus is a stationary solution to Ricci flow, i.e. .

Consider the volume of a ball of radius in the limit metric:

Now let be arbitrary.  Since , we can choose large enough that

for any .

By Glickenstein’s lemma, we have .    Applying the evolution equation for the volume form, we have the estimate

Thus we have fixed a single metric in the volume comparison limit:

Now recall that the scalar curvature is the quadratic term in the Taylor expansion for the volume of a ball:

(Here is the volume of a euclidean ball.) Letting , we get

So that in fact since was arbitrary.

On the other hand, the Bishop-Gromov comparison says that means .  So has the same ball volume as euclidean space.

This in turn implies that is flat.  So .  But each , so .  Thus we have the desired contradiction.

The smooth mean curvature flow is a family of embeddings satisfying , that is, moving by its mean curvature.  For our purposes, we will consider closed (compact without boundary).  I’ll be fairly cavalier about identifying the embedding with its image .

Now , and we may take normal coordinates so that and is a normal vector.  In these coordinates, then, .  So we may write , and think of the mean curvature flow as the heat equation for hypersurfaces.  In particular, the flow is parabolic, so we get maximum and comparison principles, as well as local existence and uniqueness.

As in the case of Ricci flow, mean curvature flow exhibits finite-time singularity development.  To see this, let be any compact hypersurface.  Then is contained in the interior of some sphere , of radius .  Now collapses with radius .  By the comparison principle, the flow starting at stays inside the region bounded by ; in particular develops a singularity before time . Sometimes, the singularity achieves is collapse to a point, as in the case of the sphere.  We have

Huisken 1984. Suppose is uniformly convex, i.e. where are the principal curvatures.  Then is isotopic to a round sphere, with the isotopy given by a rescaled flow.

What happens when we relax the convexity assumption?  We say a hypersurface is 2-convex if , where are the principal curvatures.  This is equivalent to there being at most one negative principal curvature, which is also the smallest.

Huisken-Sinestrari 2008.  Suppose is closed and 2-convex.  Then is diffeomorphic to , i.e. the boundary of a handlebody.  Moreover the flow detects the connect sums.

In particular, we’re going to need to understand what happens with two copies of connected by a long, thin tube.  It’s clear that if the tube is long and thin enough, it will collapse before either of the summands achieves a singularity.  So we’ll start by defining our singularity resolution procedure on such long, thin collapsing necks.

For our notion of neck, we’re going to use Hamilton’s idea.  A Hamilton neck is an embedding , such that the pullback metric is close to the standard round metric. (As with , I’ll blur the distinction between and its image.)  Suppose we have such a neck.

Surgery Procedure – Collapsing Neck Case.  Given a Hamilton neck, cut a region out of the middle of the neck, pinch the ends in a little, and cap them off, as in the figure.

That’s a little nontechnical, but it turns out:

Lemma.  The surgery procedure can be defined so that surgery decreases area, increases H, and increases , all by positive amounts that depend only on the scale of the neck.

Area decrease is a very coarse sort of simplification.  is a scale-invariant quantity that measures how 2-convex the surface is; we can think of convexity as being the limit case as .  Since we know how convex things behave under the flow (from Huisken’s theorem), the increase in is also a simplification of sorts. I want to claim that, in fact, this case is the only one to consider.  That is, the only singularities we see in the 2-convex case are collapsing necks.  To do this, I need to be a little more technical.  Here’s the real statement of the Huisken-Sinestrari theorem.

Surgery Theorem.  Let be 2-convex and closed.  Then there is a mean curvature flow with surgeries starting from , such that:

1. All surgeries are neck-cuttings, and all occur at the same scale .
2. All surgeries occur at times when .
3. After doing surgery, all curvatures higher than lie on components which are topological s and s.
4. There are a finite number of surgery times, after which we are left with a disjoint collection of and .

This, in turn, implies the  topological classification, since the surgery defined above is just undoing a connect sum. The theme of the proof of the Surgery Theorem is that “good surgeries suffice”.  A good surgery is:

Definition. A flow-with-surgeries is good if:

1. All surgeries are performed at the same scale .
2. Each surgery looks the same.  In particular, one side of the surgery belongs to a component that is topologically or .  The other side has a long flaring collar which starts at radius and flares to radius .
3. Each surgery is essential to disconnect the or mentioned above, which also has a point where .

The first thing to notice about good surgeries is that they can’t accumulate spatially; the long collars buffer the surgery regions away from each other spatially.  Each surgery removes the  same amount of area, and area is decreasing along the flow, so we see that there can only be finitely many surgery times.  Thus a good flow-with-surgeries has finitely many surgeries. So we can induct on the time .  In fact this is what we do:

Inductive Claim.  Suppose is good up to time and there is where .  Then we can perform a good surgery to disconnect from the rest of the manifold.

This establishes the Surgery Theorem. So far we haven’t done any singularity analysis.  In fact I’m going to relegate it to another post.  But the relevant result is:

Neck Continuation Theorem. There exist with the following property.  Suppose has and $\frac{\lambda_1}{H}(p_0,t_0)\leq \eta_1$.  Then is the centre of a neck, which can be extended in each direction until either

1. The neck closes up into a cap.
2. We reach a cross-section of radius

So if the point of high curvature we want to remove has small, we can find a cross-section with radius (radius is a continuous function which increases from about at to ); cut there.  In fact we may find either one or two such cross-sections.  In any case, the long collar exists, and the other side of the surgery is topologically recognised, so this surgery is good.

The other case, when is large, we make an end-run around, and find a nearby point where is small.  This relies on what I call the Huisken Alternative.

A trick that’s come up a lot in my recent readings is using an bound to get a bound on the supremum of some function.  The trivial case is what happens when where the constant is independent of ; then we just use the fact that the sup norm is the limit of the norms.  But what if the constant depends on , and in fact blows up as blows up?  Turns out all is not lost.

I’ll use the main theorem from Huisken 1984 as an example, though the same trick is used in Hamilton 1982 and Huisken-Sinestrari 1999.   The theorem is

Theorem. Suppose is a closed convex hypersurface.  Then there exist constants such that along a mean curvature flow starting from .

The way to prove this is to consider the function and try to prove that for some choice of , it’s bounded.  Clearly the function is bounded at each time, but we need to keep the bounds from blowing up as we approach the first singular time.

There is an important fact we’re going to use about the function .  It can absorb powers of ; that is, for any integer , we have for .  Thus if we prove a bound for , we get bounds for as well.

Now with a lot of tinkering (which is the hard part of the paper, and is different for different geometric assumptions), you can show that

Lemma.  There is a constant so that if is large and , we have .

Now we want to consider the function .  Notice that and differ only by a constant.  Using the evolution equation for , we can show without much fuss that

Now we use the Michael-Simon Sobolev inequality with and the Holder inequality to get

where all integrals are taken over the support of , i.e. the region where .  Notice that the left-hand side occurs on the right as well.

Now on the support of , so we have , where is the sup bound.  So we have

Now integrate both sides in to get

and choose so large that everywhere on .  In particular,

So we can estimate integrals of by integrals of .  Some more Holder and interpolation inequalities give

and using Holder again (stupidly this time–just pull a power of the volume out), this becomes

where is the space-time volume of the support of .  Now if , we have , so we get that

which is a sort of decay estimate on the function .  If we choose big enough that the exponent on the RHS is greater than 1, this estimate says that if is small, has to be much smaller.  So the volume is decaying pretty fast. More specifically,

Stampacchia’s Lemma. Suppose is a nonincreasing real function, which satisfies for all ,

where and .  Then there is a number with .

Thus we get a number for which , that is, for which on the entire flow.

via Ars Mathematica

Many times presuppositionalist apologists will bring up the nice crispness of logic and its unexplained utility for understanding the world as proof that God exists. There’s rationality out there, they say, which means there must be some intelligence, much like the rationality in our own heads is a product of intelligence.

I don’t find this convincing in the least, but I bring it up because today I came across an argument for the existence of God which takes the opposite tack, namely an appeal to fuzzy logic.

Fuzzy math is what you get if, instead of expecting a yes-or-no (1 or 0) answer to the question ‘Does element x belong to set X?’, you expect a number between 0 and 1. So you could answer that x .3-belongs to X.

There are a couple of interpretations available:

• the number represents a probability. That is, there is a 30% chance that x actually belongs to X. This makes fuzzy sets pretty useful in the presence of uncertainty, ala quantum mechanics, or statistics, or anything like that.
• the number represents well, fuzziness. This is useful in modeling, say, the salt concentration of an estuary.

The point that the author, Fritz Ward, wants to make is that it’s perfectly possible, in fuzzy set theory, to have a set X and an element x such that x is neither in X nor in the complement of X. Translated into logic, this says that a statement can be neither true nor false, but fall somewhere in between.

Ward wants to use this to just avoid the whole theodicy mess:

It is pleasing, in a way, that atheists can no longer rely on science to defend their position, and they are left with putting together trite syllogisms like: If God is all powerful and all knowing and all good, then evil cannot exist. But evil does exist, therefore God cannot. Implicit in this argument is the idea that the attributes some theologians (and many Christians) ascribe to God contradict, and this “proves” God does not exist. And frankly, its a pretty silly argument.

Perhaps the most fundamental misunderstanding comes by way of introducing the reader to the basic thrust of his argument:

This letter is my first attempt to fully come to grips with what the implications of fuzzy logic are, and why they offer such profound insights to people of faith. They also, incidentally, destroy traditional logic, and with it, the last remaining refuge of atheism.

There are two things wrong here.

One, it’s simply not true that fuzzy set theory (or any of the rest of topos theory) does anything like ‘destroy traditional logic’. Traditional logic works just fine, and is applicable to pretty much every situation it was applicable to before; it’s just that now we have a framework for dealing with new situations.

When people pulled their heads out of their asses about the Fifth Postulate, and realised that hyperbolic geometries worked just fine, even better, for a lot of applications, that didn’t mean all of a sudden plane (bah-dum-sh!) euclidean geometry stopped being true.

Two, there’s a majorly self-defeating flaw in Ward’s argument. He wants to argue as follows:

1. Subscribing to traditional logic is the only way to be an atheist.
2. Traditional logic has been exploded.
3. Therefore it is untenable to be an atheist.

The problem is that this argument is of the very same modus tollens form as the problem-of-evil formulation he’s arguing against. He certainly doesn’t offer any way to distinguish the two situations. He could easily, if he understands fuzzy logic so well, come up with a fuzzy-logic formulation of his argument, and prove that the problem-of-evil doesn’t have any such formulation.

But he neglects to do so, which I think is just indicative that he’s used to thinking non-fuzzily (crisply?).

What’s going on here? Another quote:

For the last century or so, beginning with Bertrand Russell, the foundations for traditional logic: in particular the bivalent claim that one cannot have both A and Not A simultaneously has crumbled. It now appears, in fact, that not only can one have both A and Not A, but that the law of the excluded middle doesn’t apply anywhere in the real world, and does not, in fact, even apply in mathematics.

Aha! Here Ward conflates the

Principle of Contradiction. Can’t have A and not-A.

with the

Law of the Excluded Middle. Either A or not-A .

There’s a huge difference here. The PC says that A and not-A are disjoint; LEM says that A and not-A cover all the possibilities. Fuzzy logic, while it allows you to say things like “.5 A”, does not allow for “A and not-A” or even “.3 A and .5 not-A”.

We also have the issue of the fact that there’s more than one fuzzy logic. Why restrict oneself to truth values between 0 and 1? Why not take truth values that are any real number? Or truth values parametrised by a projective variety? In fact people do research those logics, in a field of math called topos theory.

What’s interesting, and instructive for Dr. Ward, is that these researches are carried out within the framework of traditional logic. To prove a fuzzy logic theorem, you use ordinary logic, by constructing an ordinary-logic model of fuzzy logic. It’s again analogous to the different geometries; the Poincaré disc is a hyperbolic plane which can be embedded in the euclidean plane, so that one can use euclidean techniques to prove hyperbolic facts.

This may seem a little inelegant (it would be more satisfying to use all-hyperbolic or all-fuzzy techniques), but it works.

Recall that the elementary symmetric polynomials on are defined by

If we let denote the polynomial formed by omitting from without the terms involving . It’s immediate that

Recall also the cones . We have .

We’re going to prove

Theorem. are convex.

Proof. We’ll proceed by induction. The base case, , is clear, since is a half-space.

Now suppose that is convex. Consider the functions on , , . In fact, , i.e. is the epigraph of .
Since we’ve assumed is convex, this means that convexity of is equivalent to , where .

The relation implies

and

Dividing this last identity through by , we have

So that the function , whose concavity we’d like to establish, can be written in terms of the functions .

Now , where the bar means setting the ith component to zero. It’s easy to check that , and by our inductive hypothesis, every second derivative of is nonpositive on this set. So we have , i.e. is concave on .

The claim is now that the concavity of implies the concavity of . This follows from a computation, but you can kind of see what’s going on by noting that and are degree-1 rational functions with nonpositive second derivatives.

Consider the 1-dimensional case. A degree-1 rational function is one which is asymptotic to a line; the nonpositive second derivative means the graph of lies above the graph of the line. For example, let’s let

Then is asymptotic to the line , but since , we have that the second derivative is nonpositive everywhere. Here’s the graph of :

Now consider what happens when we take the function , analogous to . In our example, we have . This function, you may notice, is also asymptotic to , but it’s asymptotic to it from below, and its second derivative is nonnegative.

But the function we’re interested in is . Actually we’re interested in its second derivative, which means the linear term doesn’t matter, so we can discard it, and consider , which is analogous to , whose graph looks like

The relevant fact about this graph is that it has the same concavity as the original !

So concavity of , which we have by the inductive hypothesis that is convex, is enough to show that each is concave, hence is the sum of concave functions, hence itself concave. But this implies is convex, which completes the induction.

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