This month, a bunch of ‘logic hackers’ have been seeking to determine the precise boundary between the knowable and the unknowable. The challenge has been around for a long time. But only now have people taken it up with the kind of world-wide teamwork that the internet enables.

A **Turing machine** is a simple model of a computer. Imagine a machine that has some finite number of states, say N states. It’s attached to a tape, an infinitely long tape with lots of squares, with either a 0 or 1 written on each square. At each step the machine reads the number where it is. Then, based on its state and what it reads, it either *halts*, or it writes a number, changes to a new state, and moves either left or right.

The tape starts out with only 0’s on it. The machine starts in a particular ‘start’ state. It halts if it winds up in a special ‘halt’ state.

The **Busy Beaver Game** is to find the Turing machine with N states that runs as long as possible and then halts.

The number **BB(N)** is the number of steps that the winning machine takes before it halts.

In 1961, Tibor Radó introduced the Busy Beaver Game and proved that the sequence BB(N) is uncomputable. It grows faster than any computable function!

A few values of BB(N) can be computed, but there’s no way to figure out all of them.

As we increase N, the number of Turing machines we need to check increases faster than exponentially: it’s

Of course, many could be ruled out as potential winners by simple arguments. But the real problem is this: it becomes ever more complicated to determine which Turing machines with N states never halt, and which merely take a huge time to halt.

Indeed, matter what axiom system you use for math, as long as it has finitely many axioms and is consistent, you can never use it to correctly determine BB(N) for more than some finite number of cases.

So what do people know about BB(N)?

For starters, BB(0) = 0. At this point I should admit that people don’t count the halt state as one of our N states. This is just a convention. So, when we consider BB(0), we’re considering machines that only have a halt state. They instantly halt.

Next, BB(1) = 1.

Next, BB(2) = 6.

Next, BB(3) = 21. This was proved in 1965 by Tibor Radó and Shen Lin.

Next, BB(4) = 107. This was proved in 1983 by Allan Brady.

Next, BB(5). *Nobody knows what BB(5) equals!*

The current 5-state busy beaver champion was discovered by Heiner Marxen and Jürgen Buntrock in 1989. It takes 47,176,870 steps before it halts. So, we know

BB(5) ≥ 47,176,870.

People have looked at all the other 5-state Turing machines to see if any does better. But there are 43 machines that do very complicated things that nobody understands. It’s believed they never halt, but nobody has been able to prove this yet.

We may have hit the wall of ignorance here… but we *don’t know*.

That’s the spooky thing: the precise boundary between the knowable and the unknowable is unknown. It may even be unknowable… but I’m not sure we know that.

Next, BB(6). In 1996, Marxen and Buntrock showed it’s at least 8,690,333,381,690,951. In June 2010, Pavel Kropitz proved that

You may wonder how he proved this. Simple! He found a 6-state machine that runs for

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steps and then halts!

Of course, I’m just kidding when I say this was simple. The machine is easy enough to describe, but proving it takes exactly this long to run takes real work! You can read about such proofs here:

• Pascal Michel, The Busy Beaver Competition: a historical survey.

I don’t understand them very well. All I can say at this point is that many of the record-holding machines known so far are similar to the famous Collatz conjecture. The idea there is that you can start with any positive integer and keep doing two things:

• if it’s even, divide it by 2;

• if it’s odd, triple it and add 1.

The conjecture is that this process will always eventually reach the number 1. Here’s a graph of how many steps it takes, as a function of the number you start with:

Nice pattern! But this image shows how it works for numbers up to 10 million, and you’ll see it doesn’t usually take very long for them to reach 1. Usually less than 600 steps is enough!

So, to get a Turing machine that takes a long time to halt, you have to take this kind of behavior and make it much more long and drawn-out. Conversely, to analyze one of the potential winners of the Busy Beaver Game, people must take that long and drawn-out behavior and figure out a way to predict much more quickly when it will halt.

Next, BB(7). In 2014, someone who goes by the name Wythagoras showed that

It’s fun to prove lower bounds on BB(N). For example, in 1964 Milton Green constructed a sequence of Turing machines that implies

Here I’m using Knuth’s up-arrow notation, which is a recursively defined generalization of exponentiation, so for example

where there are threes in that tower.

But it’s also fun to seek the smallest N for which we can prove BB(N) is unknowable! And that’s what people are making lots of progress on right now.

Sometime in April 2016, Adam Yedidia and Scott Aaronson showed that BB(7910) cannot be determined using the widely accepted axioms for math called ZFC: that is, Zermelo—Fraenkel set theory together with the axiom of choice. It’s a great story, and you can read it here:

• Scott Aaronson, The 8000th Busy Beaver number eludes ZF set theory: new paper by Adam Yedidia and me, *Shtetl-Optimized*, 3 May 2016.

• Adam Yedidia and Scott Aaronson, A relatively small Turing machine whose behavior is independent of set theory, 13 May 2016.

Briefly, Yedidia created a new programming language, called Laconic, which lets you write programs that compile down to small Turing machines. They took an arithmetic statement created by Harvey Friedman that’s equivalent to the consistency of the usual axioms of ZFC together with a large cardinal axiom called the ‘stationary Ramsey property’, or SRP. And they created a Turing machine with 7910 states that seeks a proof of this arithmetic statement using the axioms of ZFC.

Since ZFC can’t prove its own consistency, much less its consistency when supplemented with SRP, their machine will only halt if ZFC+SRP is inconsistent.

Since most set theorists believe ZFC+SRP is consistent, this machine probably doesn’t halt. But we can’t prove this using ZFC.

In short: *if the usual axioms of set theory are consistent, we can never use them to determine the value of BB(7910)*.

The basic idea is nothing new: what’s new is the explicit and rather low value of the number 7910. Poetically speaking, we know the unknowable starts here… if not sooner.

However, this discovery set off a wave of improvements! On the Metamath newsgroup, Mario Carneiro and others started ‘logic hacking’, looking for smaller and smaller Turing machines that would only halt if ZF—that is, Zermelo–Fraenkel set theory, without the axiom of choice—is inconsistent.

By just May 15th, Stefan O’Rear seems to have brought the number down to 1919. He found a Turing machine with just 1919 states that searches for an inconsistency in the ZF axioms. Interestingly, this turned out to work better than using Harvey Friedman’s clever trick.

Thus, if O’Rear’s work is correct, we can only determine BB(1919) if we can determine whether ZF set theory is consistent. However, we cannot do this using ZF set theory—unless we find an inconsistency in ZF set theory.

For details, see:

• Stefan O’Rear, A Turing machine Metamath verifier, 15 May 2016.

I haven’t checked his work, but it’s available on GitHub.

What’s the point of all this? At present, it’s mainly just a game. However, it should have some interesting implications. It should, for example, help us better locate the ‘complexity barrier’.

I explained that idea here:

• John Baez, The complexity barrier, *Azimuth*, 28 October 2011.

Briefly, while there’s no limit on how much information a string of bits—or any finite structure—can have, there’s a limit on how much information we can *prove* it has!

This amount of information is pretty low, perhaps a few kilobytes. And I believe the new work on logic hacking can be used to estimate it more accurately!

]]>

One of the big problems with intermittent power sources like wind and solar is the difficulty of storing energy. But if we ever get a lot of electric vehicles, we’ll have a lot of batteries—and at any time, most of these vehicles are parked. So, they can be connected to the power grid.

This leads to the concept of **vehicle-to-grid** or **V2G**. In a V2G system, electric vehicles can connect to the grid, with electricity flowing from the grid to the vehicle *or back*. Cars can help solve the energy storage problem.

Here’s something I read about vehicle-to-grid systems in *Sierra* magazine:

At the University of Delaware, dozens of electric vehicles sit in a uniform row. They’re part of an experiment involving BMW, power-generating company NRG, and PJM—a regional organization that moves electricity around 13 states and the District of Columbia—that’s examining how electric vehicles can give energy back to the electricity grid.

It works like this: When the cars are idle (our vehicles typically sit 95 percent of the time), they’re plugged in and able to deliver the electricity in their batteries back to the grid. When energy demand is high, they return electricity to the grid; when demand is low, they absorb electricity. One car doesn’t offer much, but 30 of them is another story—worth about 300 kilowatts of power. Utilities will pay for this service, called “load leveling,” because it means that they don’t have to turn on backup power plants, which are usually coal or natural gas burners. And the EV owners get regular checks—approximately $2.50 a day, or about $900 a year.

It’s working well, according to Willett Kempton, a longtime V2G guru and University of Delaware professor who heads the school’s Center for Carbon-Free Power Integration: “In three years hooked up to the grid, the revenue was better than we thought. The project, which is ongoing, shows that V2G is viable. We can earn money from cars that are driven regularly.”

V2G still has some technical hurdles to overcome, but carmakers—and utilities, too—want it to happen. In a 2014 report, Edison Electric Institute, the power industry’s main trade group, called on utilities to promote EVs [electric vehicles], describing EV adoption as a “quadruple win” that would sustain electricity demand, improve customer relations, support environmental goals, and reduce utilities’ operating costs.

Utilities appear to be listening. In Virginia and North Carolina, Dominion Resources is running a pilot project to identify ways to encourage EV drivers to only charge during off-peak demand. In California, San Diego Gas & Electric will be spending $45 million on a vehicle-to-grid integration system. At least 25 utilities in 14 states are offering customers some kind of EV incentive. And it’s not just utilities—the Department of Defense is conducting V2G pilot programs at four military bases.

Paula DuPont-Kidd, a spokesperson for PJM, says V2G is especially useful for what’s called “frequency regulation service”—keeping electricity transmissions at a steady 60 cycles per second. “V2G has proven its ability to be a resource to the grid when power is aggregated,” she says. “We know it’s possible. It just hasn’t happened yet.”

I wonder how much, exactly, this system would help.

My quote comes from here:

• Jim Motavalli, Siri, will connected vehicles be greener?, *Sierra*, May–June 2016.

Motavalli also discusses vehicle-to-vehicle connectivity and vehicle-to-building systems. The latter could let your vehicle power your house during a blackout—which seems of limited use to me, but maybe I don’t get the point.

In general, it seems good to have everything I own have the ability to talk to all the rest. There will be security concerns. But as we move toward ‘ecotechnology’, our gadgets should become less obtrusive, less hungry for raw power, more communicative, and more intelligent.

]]>

A **shelf** is a set with a binary operation that distributes over itself:

There are lots of examples, the simplest being any group, where we define

They have a nice connection to knot theory, which you can see here if you think hard:

My former student Alissa Crans, who invented the term ‘shelf’, has written a lot about them, starting here:

• Alissa Crans, *Lie 2-Algebras*, Chapter 3.1: Shelves, Racks, Spindles and Quandles, Ph.D. thesis, U.C. Riverside, 2004.

I could tell you a long and entertaining story about this, including the tale of how shelves got their name. But instead I want to talk about something far more peculiar, which I understand much less well. There’s a strange connection between shelves, extremely large infinities, and extremely large finite numbers! It was first noticed by a logician named Richard Laver in the late 1980s, and it’s been developed further by Randall Dougherty.

It goes like this. For each there’s a unique shelf structure on the numbers such that

So, the elements of our shelf are

and so on, until we get to

However, we can now calculate

and so on. You should try it yourself for a simple example! You’ll need to use the self-distributive law. It’s quite an experience.

You’ll get a list of numbers, but this list will not contain all the numbers Instead, it will repeat with some period

Here is where things get weird. The numbers form this sequence:

1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, …

It may not look like it, but the numbers in this sequence approach infinity!

… *if* we assume an extra axiom, which goes beyond the usual axioms of set theory, but so far seems consistent!

This axiom asserts the existence of an absurdly large cardinal, called an I3 rank-into-rank cardinal.

I’ll say more about this kind of cardinal later. But, this is not the only case where a ‘large cardinal axiom’ has consequences for down-to-earth math, like the behavior of some sequence that you can define using simple rules.

On the other hand, Randall Dougherty has proved a *lower bound* on how far you have to go out in this sequence to reach the number 32.

And, it’s an *incomprehensibly large number!*

The third Ackermann function is roughly 2 to the th power. The fourth Ackermann function is roughly 2 raised to itself times:

And so on: each Ackermann function is defined by iterating the previous one.

Dougherty showed that for the sequence to reach 32, you have to go at least

This is an *insanely* large number!

I should emphasize that if we use just the ordinary axioms of set theory, the ZFC axioms, nobody has proved that the sequence *ever* reaches 32. Neither is it known that this is unprovable if we only use ZFC.

So, what we’ve got here is a very slowly growing sequence… which is easy to define but grows so slowly that (so far) mathematicians need new axioms of set theory to prove it goes to infinity, or even reaches 32.

I should admit that my definition of the Ackermann function is rough. In reality it’s defined like this:

And if you work this out, you’ll find it’s a bit annoying. Somehow the number 3 sneaks in:

where means raised to itself times,

where means with the number repeated times, and so on.

However, these irritating 3’s scarcely matter, since Dougherty’s number is so large… and I believe he could have gotten an even larger upper bound if he wanted.

Perhaps I’ll wrap up by saying very roughly what an I3 rank-into-rank cardinal is.

In set theory the universe of all sets is built up in stages. These stages are called the von Neumann hierarchy. The lowest stage has nothing in it:

Each successive stage is defined like this:

where is the the power set of that is, the set of all subsets of For ‘limit ordinals’, that is, ordinals that aren’t of the form we define

An I3 rank-into-rank cardinal is an ordinal such that admits a nontrivial elementary embedding into itself.

Very roughly, this means the infinity is *so huge that the collection of sets that can be built by this stage can mapped into itself, in a one-to-one but not onto way, into a smaller collection that’s indistinguishable from the original one when it comes to the validity of anything you can say about sets!*

More precisely, a nontrivial elementary embedding of into itself is a one-to-one but not onto function

that preserves and reflects the validity of all statements in the language of set theory. That is: for any sentence in the language of set theory, this statement holds for sets if and only if holds.

I don’t know why, but an I3 rank-into-rank cardinal, if it’s even consistent to assume one exists, is known to be extraordinarily big. What I mean by this is that it automatically has a lot of other properties known to characterize large cardinals. It’s inaccessible (which is big) and ineffable (which is bigger), and measurable (which is bigger), and huge (which is even bigger), and so on.

How in the world is this related to shelves?

The point is that if

are elementary embeddings, we can apply to any set in But in set theory, functions are sets too: sets of ordered pairs So, is a set. It’s not an element of but all its subsets are, where So, we can define

Laver showed that this operation distributes over itself:

And, he showed that if we take one elementary embedding and let it generate a shelf by this this operation, we get *the free shelf on one generator!*

The shelf I started out describing, the numbers with

also has one generator namely the number 1. So, it’s a quotient of the free shelf on one generator by one relation, namely the above equation.

That’s about all I understand. I don’t understand how the existence of a nontrivial elementary embedding of into itself implies that the function goes to infinity, and I don’t understand Randall Dougherty’s lower bound on how far you need to go to reach For more, read these:

• Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings, *Adv. Math.* **91** (1992), 209–231.

• Richard Laver, On the algebra of elementary embeddings of a rank into itself, *Adv. Math.* **110** (1995), 334–346.

• Randall Dougherty and Thomas Jech, Finite left distributive algebras and embedding algebras, *Adv. Math.* **130** (1997), 201–241.

• Randall Dougherty, Critical points in an algebra of elementary embeddings, *Ann. Pure Appl. Logic* **65** (1993), 211–241.

• Randall Dougherty, Critical points in an algebra of elementary embeddings, II.

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The chatter of gossip distracts us from the really big story, the Anthropocene: the new geological era we are bringing about. Here’s something that should be dominating the headlines: *Most of the Great Barrier Reef, the world’s largest coral reef system, now looks like a ghostly graveyard.*

Most corals are colonies of tiny genetically identical animals called polyps. Over centuries, their skeletons build up reefs, which are havens for many kinds of sea life. Some polyps catch their own food using stingers. But most get their food by symbiosis! They cooperate with single-celled organism called zooxanthellae. Zooxanthellae get energy from the sun’s light. They actually live inside the polyps, and provide them with food. Most of the color of a coral reef comes from these zooxanthellae.

When a polyp is stressed, the zooxanthellae living inside it may decide to leave. This can happen when the sea water gets too hot. Without its zooxanthellae, the polyp is transparent and the coral’s white skeleton is revealed—as you see here. We say the coral is bleached.

After they bleach, the polyps begin to starve. If conditions return to normal fast enough, the zooxanthellae may come back. If they don’t, the coral will die.

The Great Barrier Reef, off the northeast coast of Australia, contains over 2,900 reefs and 900 islands. It’s huge: 2,300 kilometers long, with an area of about 340,000 square kilometers. It can be seen from outer space!

With global warming, this reef has been starting to bleach. Parts of it bleached in 1998 and again in 2002. But this year, with a big El Niño pushing world temperatures to new record highs, is the worst.

Scientists have being flying over the Great Barrier Reef to study the damage, and divers have looked at some of the reefs in detail. Of the 522 reefs surveyed in the northern sector, over 80% are severely bleached and less than 1% are not bleached at all. The damage is less further south where the water is cooler—but most of the reefs are in the north:

The top expert on coral reefs in Australia, Terry Hughes, wrote:

I showed the results of aerial surveys of bleaching on the Great Barrier Reef to my students. And then we wept.

Imagine devoting your life to studying and trying to protect coral reefs, and then seeing this.

Some of the bleached reefs may recover. But as oceans continue to warm, the prospects look bleak. The last big El Niño was in 1998. With a lot of hard followup work, scientists showed that in the end, 16% of the world’s corals died in that event.

This year is quite a bit hotter.

So, global warming is not a problem for the future: it’s a problem *now*. It’s not good enough to cut carbon emissions *eventually*. We’ve got to get serious *now*.

I need to recommit myself to this. For example, I need to stop flying around to conferences. I’ve cut back, but I need to do much better. Future generations, living in the damaged world we’re creating, will not have much sympathy for our excuses.

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Biologists like Steven J. Gould like to emphasize that evolution is unpredictable. They have a point: there is absolutely no way an alien visiting the Earth 400 million years ago could have said:

Hey, I know what’s gonna happen here. Some descendants of those ugly fish will grow wings and start flying in the air. Others will walk the surface of the Earth for a few million years, but they’ll get bored and they’ll eventually go back to the oceans; when they do, they’ll be able to chat across thousands of kilometers using ultrasound. Yet others will grow arms, legs, fur, they’ll climb trees and invent BBQ, and, sooner or later, they’ll start wondering “why all this?”.

Nor can we tell if, a week from now, the flu virus will mutate, become highly pathogenic and forever remove the furry creatures from the surface of the Earth.

Evolution isn’t gravity—we can’t tell in which directions things will fall down.

One reason we can’t predict the outcomes of evolution is that genomes evolve in a super-high dimensional combinatorial space, which a ginormous number of possible turns at every step. Another is that living organisms interact with one another in a massively non-linear way, with, feedback loops, tipping points and all that jazz.

Life’s a mess, if you want my physicist’s opinion.

But that doesn’t mean that *nothing* can be predicted. Think of statistics. Nobody can predict who I’ll vote for in the next election, but it’s easy to tell what the *distribution* of votes in the country will be like. Thus, for continuous variables which arise as sums of large numbers of independent components, the central limit theorem tells us that the distribution will always be approximately normal. Or take extreme events: the max of independent random variables is distributed according to a member of a one-parameter family of so-called “extreme value distributions”: this is the content of the famous Fisher–Tippett–Gnedenko theorem.

So this is the problem I want to think about in this blog post: is evolution ruled by *statistical laws*? Or, in physics terms: does it exhibit some form of *universality*?

One lesson from statistical physics is that, to uncover universality, you need to focus on *relevant* variables. In the case of evolution, it was Darwin’s main contribution to figure out the main relevant variable: the average number of viable offspring, aka *fitness*, of an organism. Other features—physical strength, metabolic efficiency, you name it—matter only insofar as they are correlated with fitness. If we further assume that fitness is (approximately) heritable, meaning that descendants have the same fitness as their ancestors, we get a simple yet powerful dynamical principle called *natural selection*: in a given population, the lineage with the highest fitness eventually dominates, i.e. its fraction goes to one over time. This principle is very general: it applies to genes and species, but also to non-living entities such as algorithms, firms or language. The general relevance of natural selection as a evolutionary force is sometimes referred to as “Universal Darwinism”.

The general idea of natural selection is pictured below (reproduced from this paper):

It’s not hard to write down an equation which expresses natural selection in general terms. Consider an infinite population in which each lineage grows with some rate . (This rate is called the log-fitness or Malthusian fitness to contrast it with the number of viable offspring with the lifetime of a generation. It’s more convenient to use than in what follows, so we’ll just call “fitness”). Then the distribution of fitness at time satisfies the equation

whose explicit solution in terms of the initial fitness distribution

is called the **Cramér transform** of in large deviations theory. That is, viewed as a flow in the space of probability distributions, natural selection is nothing but a time-dependent exponential tilt. (These equations and the results below can be generalized to include the effect of mutations, which are critical to maintain variation in the population, but we’ll skip this here to focus on pure natural selection. See my paper referenced below for more information.)

An immediate consequence of these equations is that the mean fitness grows monotonically in time, with a rate of growth given by the variance :

The great geneticist Ronald Fisher (yes, the one in the extreme value theorem!) was very impressed with this relationship. He thought it amounted to an biological version of the second law of thermodynamics, writing in his 1930 monograph

Professor Eddington has recently remarked that “The law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of nature”. It is not a little instructive that so similar a law should hold the supreme position among the biological sciences.

Unfortunately, this excitement hasn’t been shared by the biological community, notably because this Fisher “fundamental theorem of natural selection” isn’t predictive: the mean fitness grows according to the fitness variance , but what determines the evolution of ? I can’t use the identity above to predict the speed of evolution in any sense. Geneticists say it’s “dynamically insufficient”.

But the situation isn’t as bad as it looks. The evolution of may be decomposed into the evolution of its mean , of its variance , and of its **shape** or **type**

.

(We also call the “standardized fitness distribution”.) With Ahmed Youssef we showed that:

• If is supported on the whole real line and decays at infinity as

for some , then , and converges to the standard normal distribution as . Here is the conjugate exponent to , i.e. .

• If has a finite right-end point with

for some , then , and converges to the flipped gamma distribution

Here and below the symbol means “asymptotically equivalent up to a positive multiplicative constant”; is the Heaviside step function. Note that becomes Gaussian in the limit , i.e. the attractors of cases 1 and 2 form a continuous line in the space of probability distributions; the other extreme case, , corresponds to a flipped exponential distribution.

The one-parameter family of attractors is plotted below:

These results achieve two things. First, they resolve the dynamical insufficiency of Fisher’s fundamental theorem by giving estimates of the speed of evolution in terms of the tail behavior of the initial fitness distribution. Second, they show that natural selection is indeed subject to a form of universality, whereby the relevant statistical structure turns out to be finite dimensional, with only a handful of “conserved quantities” (the and exponents) controlling the late-time behavior of natural selection. This amounts to a large reduction in complexity and, concomitantly, an enhancement of predictive power.

(For the mathematically-oriented reader, the proof of the theorems above involves two steps: first, translate the selection equation into a equation for (cumulant) generating functions; second, use a suitable Tauberian theorem—the Kasahara theorem—to relate the behavior of generating functions at large values of their arguments to the tail behavior of . Details in our paper.)

It’s useful to consider the convergence of fitness distributions to the attractors for in the skewness-kurtosis plane, i.e. in terms of the third and fourth cumulants of .

The red curve is the family of attractors, with the normal at the bottom right and the flipped exponential at the top left, and the dots correspond to numerical simulations performed with the classical Wright–Fisher model and with a simple genetic algorithm solving a linear programming problem. The attractors attract!

Statistics is useful because limit theorems (the central limit theorem, the extreme value theorem) exist. Without them, we wouldn’t be able to make any population-level prediction. Same with statistical physics: it only because matter consists of large numbers of atoms, and limit theorems hold (the H-theorem, the second law), that macroscopic physics is possible in the first place. I believe the same perspective is useful in evolutionary dynamics: it’s true that we can’t predict how many wings birds will have in ten million years, but we *can* tell what shape fitness distributions should have if natural selection is true.

I’ll close with an open question for you, the reader. In the central limit theorem as well as in the second law of thermodynamics, convergence is driven by a Lyapunov function, namely *entropy*. (In the case of the central limit theorem, it’s a relatively recent result by Arstein et al.: the entropy of the normalized sum of i.i.d. random variables, when it’s finite, is a monotonically increasing function of .) In the case of natural selection for unbounded fitness, it’s clear that entropy will also be eventually monotonically increasing—the normal is the distribution with largest entropy at fixed variance and mean.

Yet it turns out that, in our case, entropy isn’t monotonic at all times; in fact, the closer the initial distribution is to the normal distribution, the later the entropy of the standardized fitness distribution starts to increase. Or, equivalently, the closer the initial distribution to the normal, the later its relative entropy with respect to the normal. Why is this? And what’s the actual Lyapunov function for this process (i.e., what functional of the standardized fitness distribution is monotonic at all times under natural selection)?

In the plots above the blue, orange and green lines correspond respectively to

• S. J. Gould, *Wonderful Life: The Burgess Shale and the Nature of History*, W. W. Norton & Co., New York, 1989.

• M. Smerlak and A. Youssef, Limiting fitness distributions in evolutionary dynamics, 2015.

• R. A. Fisher, *The Genetical Theory of Natural Selection*, Oxford University Press, Oxford, 1930.

• S. Artstein, K. Ball, F. Barthe and A. Naor, Solution of Shannon’s problem on the monotonicity of entropy, *J. Am. Math. Soc.* **17** (2004), 975–982.

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In the triamond, each carbon atom is bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds lie in a plane, so we get a plane for each atom.

But here’s the tricky part: for any two neighboring atoms, these planes are *different.* In fact, if we draw the bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!

If we discount the difference between single and double bonds, the triamond is highly symmetrical. There’s a symmetry carrying any atom and any of its bonds to any other atom and any of *its* bonds. However, the triamond has an inherent handedness, or chirality. It comes in two mirror-image forms.

A rather surprising thing about the triamond is that the smallest rings of atoms are 10-sided. Each atom lies in 15 of these 10-sided rings.

Some chemists have argued that the triamond should be ‘metastable’ at room temperature and pressure: that is, it should last for a while but eventually turn to graphite. Diamonds are also considered metastable, though I’ve never seen anyone pull an old diamond ring from their jewelry cabinet and discover to their shock that it’s turned to graphite. The big difference is that diamonds are formed naturally under high pressure—while triamonds, it seems, are not.

Nonetheless, the mathematics behind the triamond *does* find its way into nature. A while back I told you about a minimal surface called the ‘gyroid’, which is found in many places:

• The physics of butterfly wings.

It turns out that the pattern of a gyroid is closely connected to the triamond! So, if you’re looking for a triamond-like pattern in nature, certain butterfly wings are your best bet:

• Matthias Weber, The gyroids: algorithmic geometry III, *The Inner Frame*, 23 October 2015.

Instead of trying to explain it here, I’ll refer you to the wonderful pictures at Weber’s blog.

I want to tell you a way to build the triamond. I saw it here:

• Toshikazu Sunada, Crystals that nature might miss creating, *Notices of the American Mathematical Society* **55** (2008), 208–215.

This is the paper that got people excited about the triamond, though it was discovered much earlier by the crystallographer Fritz Laves back in 1932, and Coxeter named it the Laves graph.

To build the triamond, we can start with this graph:

It’s called since it’s the complete graph on four vertices, meaning there’s one edge between each pair of vertices. The vertices correspond to four different kinds of atoms in the triamond: let’s call them red, green, yellow and blue. The edges of this graph have arrows on them, labelled with certain vectors

Let’s not worry yet about what these vectors are. What really matters is this: to move from any atom in the triamond to any of its neighbors, you move along the vector labeling the edge between them… or its negative, if you’re moving against the arrow.

For example, suppose you’re at any red atom. It has 3 nearest neighbors, which are blue, green and yellow. To move to the blue neighbor you add to your position. To move to the green one you subtract since you’re moving *against* the arrow on the edge connecting blue and green. Similarly, to go to the yellow neighbor you subtract the vector from your position.

Thus, any path along the bonds of the triamond determines a path in the graph

Conversely, if you pick an atom of some color in the triamond, any path in starting from the vertex of that color determines a path in the triamond! However, going around a loop in may not get you back to the atom you started with in the triamond.

Mathematicians summarize these facts by saying the triamond is a ‘covering space’ of the graph

Now let’s see if you can figure out those vectors.

**Puzzle 1.** Find vectors such that:

A) All these vectors have the same length.

B) The three vectors coming out of any vertex lie in a plane at 120° angles to each other:

For example, and lie in a plane at 120° angles to each other. We put in two minus signs because two arrows are pointing into the red vertex.

C) The four planes we get this way, one for each vertex, are parallel to the faces of a regular tetrahedron.

If you want, you can even add another constraint:

D) All the components of the vectors are integers.

That’s the triamond. Compare the diamond:

Here each atom of carbon is connected to four others. This pattern is found not just in carbon but also other elements in the same column of the periodic table: silicon, germanium, and tin. They all like to hook up with four neighbors.

The pattern of atoms in a diamond is called the **diamond cubic**. It’s elegant but a bit tricky. Look at it carefully!

To build it, we start by putting an atom at each *corner* of a cube. Then we put an atom in the middle of each *face* of the cube. If we stopped there, we would have a **face-centered cubic**. But there are also four more carbons inside the cube—one at the center of each tetrahedron we’ve created.

If you look really carefully, you can see that the full pattern consists of two interpenetrating face-centered cubic lattices, one offset relative to the other along the cube’s main diagonal.

The face-centered cubic is the 3-dimensional version of a pattern that exists in any dimension: the **D _{n} lattice**. To build this, take an n-dimensional checkerboard and alternately color the hypercubes red and black. Then, put a point in the center of each black hypercube!

You can also get the D_{n} lattice by taking all n-tuples of integers that sum to an even integer. Requiring that they sum to something *even* is a way to pick out the black hypercubes.

The diamond is also an example of a pattern that exists in any dimension! I’ll call this the **hyperdiamond**, but mathematicians call it **D _{n}^{+}**, because it’s the union of two copies of the D

In any dimension, the volume of the unit cell of the hyperdiamond is 1, so mathematicians say it’s **unimodular**. But only in even dimensions is the sum or difference of any two points in the hyperdiamond again a point in the hyperdiamond. Mathematicians call a discrete set of points with this property a **lattice**.

If even dimensions are better than odd ones, how about dimensions that are multiples of 4? Then the hyperdiamond is better still: it’s an **integral** lattice, meaning that the dot product of any two vectors in the lattice is again an integer.

And in dimensions that are multiples of 8, the hyperdiamond is even better. It’s **even**, meaning that the dot product of any vector with itself is even.

In fact, even unimodular lattices are only possible in Euclidean space when the dimension is a multiple of 8. In 8 dimensions, the only even unimodular lattice is the 8-dimensional hyperdiamond, which is usually called the **E _{8} lattice**. The E

To me, the glittering beauty of diamonds is just a tiny hint of the overwhelming beauty of E_{8}.

But let’s go back down to 3 dimensions. I’d like to describe the diamond rather explicitly, so we can see how a slight change produces the triamond.

It will be less stressful if we double the size of our diamond. So, let’s start with a face-centered cubic consisting of points whose coordinates are even integers summing to a multiple of 4. That consists of these points:

(0,0,0) (2,2,0) (2,0,2) (0,2,2)

and all points obtained from these by adding multiples of 4 to any of the coordinates. To get the diamond, we take all these together with another face-centered cubic that’s been shifted by (1,1,1). That consists of these points:

(1,1,1) (3,3,1) (3,1,3) (1,3,3)

and all points obtained by adding multiples of 4 to any of the coordinates.

The triamond is similar! Now we start with these points

(0,0,0) (1,2,3) (2,3,1) (3,1,2)

and all the points obtain from these by adding multiples of 4 to any of the coordinates. To get the triamond, we take all these together with another copy of these points that’s been shifted by (2,2,2). That other copy consists of these points:

(2,2,2) (3,0,1) (0,1,3) (1,3,0)

and all points obtained by adding multiples of 4 to any of the coordinates.

Unlike the diamond, the triamond has an inherent handedness, or chirality. You’ll note how we used the point (1,2,3) and took cyclic permutations of its coordinates to get more points. If we’d started with (3,2,1) we would have gotten the other, mirror-image version of the triamond.

I mentioned that the triamond is a ‘covering space’ of the graph More precisely, there’s a graph whose vertices are the atoms of the triamond, and whose edges are the bonds of the triamond. There’s a map of graphs

This automatically means that every path in is mapped to a path in But what makes a **covering space** of is that any path in comes from a path in which is *unique* after we choose its starting point.

If you’re a high-powered mathematician you might wonder if is the universal covering space of It’s not, but it’s the universal *abelian* covering space.

What does this mean? Any path in gives a sequence of vectors and their negatives. If we pick a starting point in the triamond, this sequence describes a unique path in the triamond. *When does this path get you back where you started?* The answer, I believe, is this: if and only if you can take your sequence, rewrite it using the commutative law, and cancel like terms to get zero. This is related to how adding vectors in is a commutative operation.

For example, there’s a loop in that goes “red, blue, green, red”. This gives the sequence of vectors

We can turn this into an expression

However, we can’t simplify this to zero using just the commutative law and cancelling like terms. So, if we start at some red atom in the triamond and take the unique path that goes “red, blue, green, red”, we do not get back where we started!

Note that in this simplification process, we’re not allowed to use what the vectors “really are”. It’s a purely formal manipulation.

**Puzzle 2.** Describe a loop of length 10 in the triamond using this method. Check that you can simplify the corresponding expression to zero using the rules I described.

A similar story works for the diamond, but starting with a different graph:

The graph formed by a diamond’s atoms and the edges between them is the universal abelian cover of this little graph! This graph has 2 vertices because there are 2 kinds of atom in the diamond. It has 4 edges because each atom has 4 nearest neighbors.

**Puzzle 3.** What vectors should we use to label the edges of this graph, so that the vectors coming out of any vertex describe how to move from that kind of atom in the diamond to its 4 nearest neighbors?

There’s also a similar story for graphene, which is hexagonal array of carbon atoms in a plane:

**Puzzle 4.** What graph with edges labelled by vectors in should we use to describe graphene?

I don’t know much about how this universal abelian cover trick generalizes to higher dimensions, though it’s easy to handle the case of a cubical lattice in any dimension.

**Puzzle 5.** I described higher-dimensional analogues of diamonds: are there higher-dimensional triamonds?

The Wikipedia article is good:

• Wikipedia, Laves graph.

They say this graph has many names: the **K _{4} crystal**, the

This paper describes various attempts to find the Laves graph in chemistry:

• Stephen T. Hyde, Michael O’Keeffe, and Davide M. Proserpio, A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics, *Angew. Chem. Int. Ed.* **47** (2008), 7996–8000.

This paper does some calculations arguing that the triamond is a metastable form of carbon:

• Masahiro Itoh *et al*, New metallic carbon crystal, *Phys. Rev. Lett.* **102** (2009), 055703.

Abstract.Recently, mathematical analysis clarified that sp^{2}hybridized carbon should have a three-dimensional crystal structure () which can be regarded as a twin of the sp^{3}diamond crystal. In this study, various physical properties of the carbon crystal, especially for the electronic properties, were evaluated by first principles calculations. Although the crystal is in a metastable state, a possible pressure induced structural phase transition from graphite to was suggested. Twisted π states across the Fermi level result in metallic properties in a new carbon crystal.

The picture of the crystal was placed on Wikicommons by someone named ‘Workbit’, under a Creative Commons Attribution-Share Alike 4.0 International license. The picture of the tetrahedron was made using Robert Webb’s Stella software and placed on Wikicommons. The pictures of graphs come from Sunada’s paper, though I modified the picture of The moving image of the diamond cubic was created by H.K.D.H. Bhadeshia and put into the public domain on Wikicommons. The picture of graphene was drawn by Dr. Thomas Szkopek and put into the public domain on Wikicommons.

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Here’s a new one:

• Joel David Hamkins, Any function can be computable.

Let me try to explain it without assuming you’re an expert on mathematical logic. That may be hard, but I’ll give it a try. We need to start with some background.

First, you need to know that there are many different ‘models’ of arithmetic. If you write down the usual axioms for the natural numbers, the Peano axioms (or ‘PA’ for short), you can then look around for different structures that obey these axioms. These are called ‘models’ of PA.

One of them is what *you* think the natural numbers are. For you, the natural numbers are just 0, 1, 2, 3, …, with the usual way of adding and multiplying them. This is usually called the ‘standard model’ of PA. The numbers 0, 1, 2, 3, … are called the ‘standard’ natural numbers.

But there are also nonstandard models of arithmetic. These models contain extra numbers beside the standard ones! These are called ‘nonstandard’ natural numbers.

This takes a while to get used to. There are several layers of understanding to pass through.

For starters, you should think of these extra ‘nonstandard’ natural numbers as bigger than all the standard ones. So, imagine a whole bunch of extra numbers tacked on after the standard natural numbers, with the operations of addition and multiplication cleverly defined in such a way that all the usual axioms still hold.

You can’t just tack on finitely many extra numbers and get this to work. But there can be countably many, or uncountably many. There are infinitely many different ways to do this. They are all rather hard to describe.

To get a handle on them, it helps to realize this. Suppose you have a statement S in arithmetic that is neither provable nor disprovable from PA. Then S will hold in some models of arithmetic, while its negation not(S) will hold in some other models.

For example, the Gödel sentence G says “this sentence is not provable in PA”. If Peano arithmetic is consistent, neither G nor not(G) is provable in PA. So G holds in some models, while not(G) holds in others.

Thus, you can intuitively think of different models as “possible worlds”. If you have an undecidable statement, meaning one that you can’t prove or disprove in PA, then it holds in some worlds, while its negation holds in other worlds.

In the case of the Gödel sentence G, most mathematicians think G is “true”. Why the quotes? Truth is a slippery concept in logic—there’s no precise definition of what it means for a sentence in arithmetic to be “true”. All we can precisely define is:

1) whether or not a sentence is provable from some axioms

and

2) whether or not a sentence holds in some model.

Nonetheless, mathematicians are human, so they have beliefs about what’s true. Many mathematicians believe that G is true: indeed, in popular accounts one often hears that G is “true but unprovable in Peano arithmetic”. So, these mathematicians are inclined to say that any model where G doesn’t hold is nonstandard.

Anyway, what is Joel David Hamkins’ result? It’s this:

There is a Turing machine T with the following property. For any function from the natural numbers to the natural numbers, there is a model of PA such that

in this model, if we give T any standard natural as input, it halts and outputs

So, take to be your favorite uncomputable function. Then there’s a model of arithmetic such that *in this model*, the Turing machine computes at least when you feed the machine standard numbers as inputs.

So, *very very* roughly, there’s a possible world in which your uncomputable function becomes computable!

But you have to be very careful about how you interpret this result.

What’s the trick? The proof is beautiful, but it would take real work to improve on Hamkins’ blog article, so please read that. I’ll just say that he makes extensive use of Rosser sentences, which say:

“For any proof of this sentence in theory T, there is a smaller proof of the negation of this sentence.”

Rosser sentences are already mind-blowing, but Hamkins uses an *infinite sequence* of such sentences and their negations, chosen in a way that depends on the function to cleverly craft a model of arithmetic in which the Turing machine T computes this function on standard inputs.

But what’s really going on? How can using a nonstandard model make an uncomputable function become computable for standard natural numbers? Shouldn’t nonstandard models agree with the standard one on this issue? After all, the only difference is that they have extra nonstandard numbers tacked on after all the standard ones! How can that make a Turing machine succeed in computing on *standard* natural numbers?

I’m not 100% sure, but I think I know the answer. I hope some logicians will correct me if I’m wrong.

You have to read the result rather carefully:

There is a Turing machine T with the following property. For any function from the natural numbers to the natural numbers, there is a model of PA such that

in this model, if we give T any standard natural as input, it halts and computes

When we say the Turing machine halts, we mean it halts after steps for some natural number But this may not be a standard natural number! It’s a natural number in the model we’re talking about.

So, the Turing machine halts… but perhaps only after a nonstandard number of steps.

In short: you can compute the uncomputable, but only if you’re willing to wait long enough. *You may need to wait a nonstandard amount of time*.

It’s like that old Navy saying:

But the trick becomes more evident if you notice that *one single* Turing machine T computes *different functions* from the natural numbers to the natural numbers… in different models. That’s even weirder than computing an uncomputable function.

The only way to build a machine that computes in one model and in another to build a machine that doesn’t halt in a standard amount of time in either model. It only halts after a *nonstandard* amount of time. In one model, it halts and outputs In another, it halts and outputs

To dig a bit deeper—and this is where it gets a bit scary—we have to admit that the standard model is a somewhat elusive thing. I certainly didn’t define it when I said this:

For you, the natural numbers are just 0, 1, 2, 3, …, with the usual way of adding and multiplying them. This is usually called the

standard modelof PA. The numbers 0, 1, 2, 3, … are called the ‘standard’ natural numbers.

The point is that “0, 1, 2, 3, …” here is vague. It makes sense if you already know what the standard natural numbers are. But if you don’t already know, those three dots aren’t going to tell you!

You might say the standard natural numbers are those of the form 1 + ··· + 1, where we add 1 to itself some finite number of times. But what does ‘finite number’ mean here? It means a standard natural number! So this is circular.

So, conceivably, the concept of ‘standard’ natural number, and the concept of ‘standard’ model of PA, are more subjective than most mathematicians think. Perhaps some of my ‘standard’ natural numbers are nonstandard for you!

I think most mathematicians would reject this possibility… but not all. Edward Nelson tackled it head-on in his marvelous book *Internal Set Theory*. He writes:

Perhaps it is fair to say that “finite” does not mean what we have always thought it to mean. What have we always thought it to mean? I used to think that I knew what I had always thought it to mean, but I no longer think so.

If we go down this road, Hamkins’ result takes on a different significance. It says that any subjectivity in the notion of ‘natural number’ may also infect what it means for a Turing machine to halt, and what function a Turing machine computes when it does halt.

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Here you can see the brilliant flash of a supernova as its core blasts through its surface. This is an animated cartoon made by NASA based on observations of a red supergiant star that exploded in 2011. It has been sped up by a factor of 240. You can see a graph of brightness showing the actual timescale at lower right.

When a star like this runs out of fuel for nuclear fusion, its core cools. That makes the pressure drop—so the core collapses under the force of gravity.

When the core of a supernova collapses, the infalling matter can reach almost a quarter the speed of light. So when it hits the center, this matter becomes *very* hot! Indeed, the temperature can reach 100 billion kelvin. That’s 6000 times the temperature of our Sun’s core!

For a supernova less than 25 solar masses, the collapse stops only when the core is compressed into a neutron star. As this happens, lots of electrons and protons become neutrons and neutrinos. Most of the resulting energy is instantly carried away by a ten-second burst of neutrinos. This burst can have an energy of 10^{46} joules.

It’s hard to comprehend this. It’s what you’d get if you suddenly converted the mass of 18,000 Earths into energy! Astronomers use a specially huge unit with such energies: the **foe**, which stands for ten to the **f**ifty-**o**ne **e**rgs.

That’s 10^{44} joules. So, a supernova can release 100 foe in neutrinos. By comparison, only 1 or 2 foe come out as light.

Why? Neutrinos can effortlessly breeze through matter. Light cannot! So it takes longer to actually *see* things happen at the star’s surface—especially since a red supergiant is *large*. This one was about 500 times the radius of our Sun.

So what happened? A shock wave rushed upward through the star. First it broke through the star’s surface in the form of finger-like plasma jets, which you can see in the animation.

20 minutes later, the full fury of the shock wave reached the surface—and the doomed star exploded in a blinding flash! This is called the **shock breakout**.

Then the star expanded as a blue-hot ball of plasma.

Here’s how the star’s luminosity changed with time, measured in multiples of the Sun’s luminosity:

Note that while the shock breakout seems very bright, it’s ultimately dwarfed by the luminosity of the expanding ball of plasma. So, KSN2011d was actually one of the first two supernovae for which the shock breakout was seen! For details, read this:

• P. M. Garnavich, B. E. Tucker, A. Rest, E. J. Shaya, R. P. Olling, D. Kasen and A. Villar, Shock breakout and early light curves of Type II-P supernovae observed with Kepler.

A **Type II** supernova is one that shows hydrogen in its spectral lines: these are commonly formed by the collapse of a star that has run out of fuel in its core, but retains hydrogen in its outer layers. A **Type II-P** is one that shows a plateau in its light curve: the P is for ‘plateau’. These are more common than the **Type II-L**, which show a more rapid (‘linear’) decay in their luminosity:

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Traditional Tom and Liberal Lisa are dating. They discuss their plans for having children:

**Tom:** I plan to keep having kids until I get two sons in a row.

**Lisa:** What?! That’s absurd. Why?

**Tom:** I want two to run my store when I get old.

**Lisa:** Even ignoring your insulting assumption that only *boys* can manage your shop, why in the world do you need *two in a row?*

**Tom:** From my own childhood, I’ve learned there’s a special bond between sons who are next to each other in age. They play together, they grow up together… they can run my shop together.

**Lisa:** Hmm. Well, then maybe I should have children until I have a girl followed directly by a boy!

**Tom:** What?!

**Lisa:** Well, I’ve observed that something special happens when a boy has an older sister, with no intervening siblings. They play together, they grow up together… and maybe he learns not to be such a sexist pig!

They decide they are incompatible, so they split up and each one separately tries to find a mate who will go along with their reproductive plan.

Now for some puzzles. In these puzzles, assume that each time someone has a child, they have a 50% chance of having either a daughter or a son. Also assume each event is independent: that is, the gender of any children so far has no effect on that of later ones. Also ignore twins and other tricky issues.

**Puzzle 1.** If Tom carries out his plan of having children until he has two consecutive sons, and then stops, what is the expected number of children he will have?

**Puzzle 2.** If Lisa carries out her plan of having children until she has a daughter followed directly by a son, and then stops, what is the expected number of children she will have?

**Puzzle 3:** Which is greater, Tom’s expected number of children or Lisa’s? Or are they equal?

For maximum benefit, try to answer Puzzle 3 before doing the calculations required to answer Puzzles 1 or 2.

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About a year ago, the International Energy Agency announced some important news. Although the global GDP grew by 3.4% in 2014, greenhouse gas emissions due to energy use did not increase! We spewed 32.3 gigtonnes of carbon dioxide into the atmosphere by burning stuff to produce energy—just as we had in 2013.

Of course, leveling off is not good enough. Since carbon dioxide stays in the atmosphere essentially ‘forever’, we need to essentially *quit* burning stuff. You can’t stop a clogged sink from overflowing by *levelling off* the rate at which you pour in water. You have to *turn off the faucet!*

But still, it’s a promising start.

And now the IEA is saying the same thing about 2015. While the global GDP grew 3.1% in 2015, we spewed just 32.1 billion gigatonnes of CO_{2} into the air by burning stuff to make energy. So these carbon emissions are flat or even slightly down from 2014!

The IEA put out a press release about this:

• International Energy Agency, Decoupling of global emissions and economic growth confirmed, 16 March 2016.

and here is some of what it says:

“The new figures confirm last year’s surprising but welcome news: we now have seen two straight years of greenhouse gas emissions decoupling from economic growth,” said IEA Executive Director Fatih Birol. “Coming just a few months after the landmark COP21 agreement in Paris, this is yet another boost to the global fight against climate change.”

Global emissions of carbon dioxide stood at 32.1 billion tonnes in 2015, having remained essentially flat since 2013. The IEA preliminary data suggest that electricity generated by renewables played a critical role, having accounted for around 90% of new electricity generation in 2015; wind alone produced more than half of new electricity generation. In parallel, the global economy continued to grow by more than 3%, offering further evidence that the link between economic growth and emissions growth is weakening.

In the more than 40 years in which the IEA has been providing information on CO

_{2}emissions, there have been only four periods in which emissions stood still or fell compared to the previous year. Three of those—the early 1980s, 1992 and 2009—were associated with global economic weakness. But the recent stall in emissions comes amid economic expansion: according to the International Monetary Fund, global GDP grew by 3.4% in 2014 and 3.1% in 2015.The two largest emitters, China and the United States, both registered a decline in energy-related CO

_{2}in 2015. In China, emissions declined by 1.5%, as coal use dropped for the second year in a row. The economic restructuring towards less energy-intensive industries and the government’s efforts to decarbonise electricity generation pushed coal use down. In 2015, coal generated less than 70% of Chinese electricity, ten percentage points less than four years ago (in 2011). Over the same period low-carbon sources jumped from 19% to 28%, with hydro and wind accounting for most of the increase. In the United States, emissions declined by 2%, as a large switch from coal to natural gas use in electricity generation took place.The decline observed in the two major emitters was offset by increasing emissions in most other Asian developing economies and the Middle East, and also a moderate increase in Europe.

More details on the data and analysis will be included in a World Energy Outlook special report on energy and air quality that will be released at the end of June. The report will go beyond CO

_{2}emissions and will provide a first in-depth analysis of the role the energy sector plays in air pollution, a crucial policy issue that today results in 7 million premature deaths a year. The report will provide the outlook for emissions and their impact on health, and provide policy makers with strategies to mitigate energy-related air pollution in the short and long term.To download annual energy-related CO

_{2}emissions data, click here.To read last year’s announcement about CO

_{2}emissions, click here.

Here’s an optimistic assessment of what’s been going since the Paris Agreement was sealed on 12 December 2015:

• Paris Agreement 100 days on: The dawn of a new era?, *BusinessGreen*, 21 March 2016.

It’s mainly interesting to me because it has a passage with lots of links. I’ll quote that part:

Just days after the agreement, the Obama administration pulled off another coup extending renewable energy tax credits and effectively engineering an acceleration of the country’s renewable energy boom. China followed a few months later with a Five Year Plan that majored on environmental progress and further fuelled speculation the superpower’s coal use has already peaked. Canada continued its rehabilitation from climate villain to climate champion, inking a comprehensive bilateral agreement with the US to crackdown on methane emissions and put another stake through the heart of Arctic drilling plans. Sweden edged forward with plans for a carbon neutral economy, as Japan revealed plans to accelerate its emission reductions through to 2030. And the UK government, sadly still a byword for climate policy contrariness, revealed it would take the over-arching goal of the Paris Agreement and enshrine it in national law through a new target to build a net zero emission economy.

This global policy push, coupled with inexorable technological progress (witness the latest record-breaking solar cells and the blistering pace of improvements in energy storage technology), is working. Just weeks after the Paris Agreement the clean energy investment and greenhouse gas emission data for 2015 started to come in, and the stats were better than anyone could have expected. Clean energy investment reached a record $329bn, as it became increasingly clear renewables are now the generation option of choice in multiple markets around the world. In industrialised countries such as the UK emissions kept falling fast, while the IEA suggested emissions globally are remaining flat, despite increasing wealth.

These mega trends are inevitably being felt at the coal face, so to speak, of modern business. Since Paris, US coal giant Arch Coal filed for bankruptcy and Peabody Energy warned it may have to do the same. In the UK, mainstream energy trade body Energy UK delivered its own Road to Damascus moment, announcing its members were primed and ready to deliver a low carbon transition. Iberdrola, one of the few European utilities closely associated with a full bore commitment to decarbonisation, became one of the few European utilities to report decent financial results. The march of the divestment movement continued, as savvy investors all over the world have internalised the logic of the Paris Agreement’s goals and recognised that carbon intensive business models’ days are numbered. The flight from high risk coal assets gathered pace, just as the development of high risk oil assets slowed.

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